I did at first, and now I can mimic the recordings of the sentences for HebrewPod's Hebrew Word of the Day quite well. The secret is to do one syllable at a time, and try to think of a context in which you might make a similar sound in English. For instance, you might say "ba'a" with the little stop in the syllables if you got interrupted by someone, or jolted over a bump while riding on a bicycle.
Using the "i" or the "ee" depends on how you want to define them. In reading English, frequently an "i" has the sound as mentioned, in "hit". This is NOT a sound in Hebrew. However, the "i" as it is used when writing in other languages (such as Romaji), where it is pronounced like a long "e" in English, is exactly the right sound in this case. The only way to always write long "e" in English, without fancy characters, is "ee". So I think both ways of writing can be correct. To conclude, the word היא would be pronounced like the English word "he." I agree on the "ba'a" part.
In Hebrew there used to be distinction between long and short vowels. Nowadays, they are all short, so you saying there is no short i in Hebrew is not correct. English still makes that distinction. For me "ee" denotes a long vowel and "i" is short. Therefore, only "i" is correct when transcribing. The same goes for other vowels. A, E, O, U and not AH, EH, OH, OO, as to me they all point to a long vowel sound.