How could a time traveller change medieval history using only modern knowledge?

Trying to implement modern tehnology in a short time is a non-starter. Don't go that way. Use the local tehnology and military knowledge augmented with the unique intelligence opportunity brought by the special advisor.

They are a time traveller, right? So they know (1) how large is the opposing force and its composition, (2) when it will be assembled and where, (3) what route they will take and (4) where the battle is to be fought. Using this knowledge they can instruct the king to:

• Assemble a larger force. Medieval armies were tiny. Any half decent kingdom could easily assemble a larger force than what the enemy thought was enough -- in medieval times the real problem was the abysmal logistics, so, if the king knows precisely when and were to assemble to host it should be easily feasible to achieve superior numbers.

• Counteract the composition of the opposing force. The enemy comes with heavy cavalry? Prepare the field with caltrops, masked ditches, and other anti-cavalry surprises. If they have a little time they could introduce effective anti-cavalry tactics, such as pike formations. The enemy relies on Genoese mercenaries armed with cross-bows? Fight behind field fortifications. Surely the generals will know what to do if you tell them in advance what the enemy will bring.

• Choose a better battlefield. The time traveller knows the route and the schedule of the opposing force: the generals of the king can exploit this knowledge to choose a battlefield which favors them.

• By knowing the size, composition, route and schedule of the opposing force your side can easily achieve strategic surprise; if they can't they amply deserve to lose. Launch a pre-emptive attack. Attach the opposing force en route. Defeat the opponents in detail before they can concentrate their forces.

Some ideas:

Steam machines: Are not that difficult to build for an experienced blacksmith. You may get ironclads and other steam machines.

Distiled Alcohol: It's highly flammable a not so difficult to get from wine or other alcoholic beverages. You can create plenty of machines being fueled by alcohol.

Optics: Not that difficult if you have people who know how to make glass works. You can get binoculars and lenses.

Hot Air Balloons: Not difficult to make. They would give archers some advantage and would frighten your enemies.

Compressed Air Cannons and Catapults: Once you know how to build valves and gears is not that difficult.

Modern medicine will help a lot, reducing the number of people dying from wounds. I am talking about knowing germs exists and being able to cultivate penicillin, with that you could reduce the number of deaths greatly. Hygiene will keep your soldiers free of illnesses while knowing the importance of vitamins (found around renaissance era) will keep them strong.

Steam power will also help to power catapults and similar siege weaponry. Increasing their power and reducing the number of people manning them.

Let's assume our TT have no knowledge of this battle so he can't prepare and predicts for particular moves.
So chemistry - gunpowder is safe and easy choice, there is such things as "difficult ingredients", it's just coal, sulphur and potassium nitrate that had to be imported from India anyway. Then comes to mind hydrochloric acid or sulphur acid. You don't need much to have a great psychological impact on the enemy. Second one is mustard gas (because if you have hydrochloric acid you can make mustard gas).
But that's just dirty tricks. If you want to play clean then:

• better mining equipment for trenches and pits
• entanglements
• artillery (balistas made with modern physics and more efficient ammo)
• Smaller army means better equipment for fewer people is needed, better armour with more movability, better training and more diverse use of cavalry.

This is kind of broad, but anybody with sufficient modern scientific knowledge and modern combat tactics can potentially determine the outcome of any battle in the medieval times.

While the possibilities are almost endless, here are some which I thought would be the easiest and the most effective.

1- Use historical knowledge and battle specifics

Someone from our times would know the details about the particular battle and how the armies were arranged, which weapons were used and what were the strengths and weaknesses of each side. With this hindsight knowledge, you can easily overcome your party's weaknesses and focus on the weaknesses of your foe. Knowledge is power, and future knowledge is like nuclear power!

2- Utilize biochemical weapons to decimate and demoralize the enemy

You have lots and lots and lots of options here. Coat your arrows and spears with venom collected from snakes, scorpions, spiders, poison dart frogs (central and South America only) and hornets. For mineral toxins to coat your blades, use corrosive sublimate (HgCl$_2$), phosphorous salts and sulfide salts. In plant poisons, you can use oleander, calotropis, digitalis, bushman's poison, strychnos, datura, rosary pea, wolfsbane, hemlock ... (and a couple dozen others. I think the point has been made)

3- Utilize wind!

Before the battle, when the armies are camped in front of each other, if the wind is blowing from your side to them, burn noxious chemicals so that the fumes are blown toward your enemy, causing damage and panic. Large torches, coated with a mixture of crude oil, sulfur and powdered oleander will do the job nicely. The fumes will be acidic and toxic, not only inducing cough and teary eyes, but also causing panic and decrease the morale of the enemy.

4- Use blinding beam to disorient your enemy

You would need many mirrors of 1 square foot size. Install the army of these mirror holders (about 300 of these will do) at some out-of-reach location (e.g. at the top of a very steep hill). They will focus the reflected sunlight beam at the enemy's soldiers (specifically targetting archers and cavalry). While they will not be able to set anyone on fire or inflict severe burns, they will be able to blind them and create panic and disorientation, helping your side.

5- Implant poison coated punji sticks in the way of your enemy

I mean these.

Considering that you (the time traveller) in hindsight, knows which path the enemy will take to the battlefield, you can easily deploy tens of thousands of these pointed sticks with poisoned tips, in his way. You can plant them in the jungles and woods in his path and you can lay punji traps for him, by digging up pits, planting punji sticks in them and then covering up the top with a thin layer of grass and sticks. Given a couple weeks, the upper layer would completely naturalize with the earth, making it impossible to see the trap.

The results would be simply disastrous for the enemy.

6- Use caltrops to disable cavalry

These are caltrops.

Prepare lots of these and give at least a couple dozen to each foot soldier and cavalryman in your army. Utilize them to disable fast cavalry attacks during the battle. Simply throw the caltrops in the way of the invading cavalry and many of them will never reach you at all. It will also slow down the approach of the remainders, as they will have to watch out for these deadly spikes.

Final Word

There are many, many more methods you can deploy, but I thought these would be the easiest and most potent ones. The medic improvement and steam powered catapults (which I don't think can be built during a 6 months time) have already been mentioned by other users, so I do not need to reiterate any of those.

Depending on how technically savvy this person is, the time traveler would actually be able to create a stream engine and kick start the industrial revolution by simply introducing the people to this technology. He could also introduce electricity and make generators driven by the previously introduced steam engine. With enhanced industry the production of armor and weapons can increase granted they manage to set up production soon enough.

With the introduction of the steam engine new ways to conduct agriculture also emerge. With this new technology they can easily establish a way more efficient resource production and likely higher quality armor and weapons, thus overwhelming the opponent.

On an even longer time, with the technological kickstart the kingdom received it will soon surpass technology and wealth of every bordering kingdom.

Politics.

If you came up of a really good invention - you will still have to build it using the medieval materials, and then produce it in big numbers, and then train soldiers to use it.
If you only have a year, you probably don't have the time for that.

But you got the ear of the king (umm, I hope not literally), so you can at least win more time, or maybe even avoid the var.
You probably know the situation in the opposed kingdom. Maybe you'll be able to find a common enemy, or to spark a rebellion there, or to provoke it to attack some other country first. Maybe try to appear more threatening (showing off steam engines?), or more useful as an ally (penicillin trade pact?).

But in the meantime, yes, keep improving mining, medicine and hygiene. You probably will be attacked by someone else, anyway.