Я помогу + написать or писать?

I was expecting an imperfective verb, because "writing" is a process.

Perfective verbs are about state transition and imperfective verbs are about state.

Usually (usually) the state implies some process, and the state transition implies the start or the end of this process, but it's not always the case.

Я помогу Татьяне писать письмо

This would mean

I will help Tatiana to stay in her "letter is being written" mode.

The state here is "the letter is being written".

This is something you would say, for instance, when the letter is 50 pages long and it takes more than one day to write it.

You don't necessarily mean that the letter will be finished after you're done helping. What you're saying is that you'll see to it that she has everything she needs to keep pushing towards the end.

Я помогу Татьяне написать письмо

This would mean

I will help Tatiana to get from the world where the letter is not done to the world where the letter is done.

The state transition happens when she has dotted all her i's and crossed all her t's and the letter is ready to be sealed and mailed.

You are saying that you'll see to it that Tatiana has everything she needs to make this transition happen and you will keep helping her until it really happens.