People would probably interpret your sentence as meaning she sent a letter to her own grandmother, but you said you’re just learning grammar, so presumably they are trying to be as explicit as possible.
Just be aware that not specifying can result in different interpretations based on the prior context of the conversation.
I have a book: 501 Japanese Verbs
I’ve also been looking at this site: Japanese Verb Conjugator
(I know it’s an auto-program that may not be accurate)
I noticed neither of these resources have the たい form of the verbs:
買いたい - Want to buy
I also tried Jisho.org to see if it was in the Inflections section, but it wasn’t.
What is this form usually called in English (e.g. presumptive, imperative, etc)?
And any idea why they don’t list it?
Interestingly the word I picked at random for an example in this post (買う) had someone ask about it in the comments section for that word at the bottom of the page
Yeah, the fact that they’re adjectives seems like a reason they wouldn’t be grouped with verb conjugations. Just like how explanations of adding の or こと to verbs to make nouns would be separate.
To add to this, basically for compound words, if the second word starts with the same letter as the first word ends or follows a つ, but does not have dakuten, then the sound merges.
To the best of my knowledge, you will never see ヶ again outside of counting months.
But for your purposes both か and ヶ are correct although 六ヶ月 is the correct “kanji” way of writing it, both appear in the wild.