How do I type “party”? I can’t get that “te-small-i” pair to appear. Just “chi-small-i”.

I believe you can do it this way:
pa-teli-
Which would translate to:
パーテイー
By typing “li” it creates the small イ symbol.
On Microsoft IME, you need to type xi to get ぃ/ィ.
You wrote パーチイー instead of パーテイー, so in other words “paachii” vs “paatii”
Thank you so much!