Question: I was trying to transcribe my cat's noises into IPA. For example, her wanting to be petted is like [njæː] or [njaː]. But purring doesn't seem to fit any existing notation, except maybe [ʜ]. Has anyone ever mapped a cat's phonetic inventory? Their vocal tracts don't seem to be much different than ours.

Answer: 
Aside from @user6726's answer this is my personal view:
Phonetic transcription is based on human's articulation (as human's produce for this case you can say for example the /k/ sound is palatal but do cats use hard palate to produce sounds?!) Imagine a hypothetical creature that makes a sound like modem's `(actually old dial-up modems)` sound. we would transcribe that sound in a way that our mind hears it like it's language. for better understanding you can have this in mind that in languages like Persian the cat's meow is transcribed as `/miu:/` (not sure if I transcribed it correctly but it's long /u/) having said that you first have to see how the cat articulates this sound (you could use MRI or cameras) then you have to see how the cat produces it and then you can assign a symbol for it.

**BUT** there is another assumption in question: Can we use IPA for transcribing cat's sounds? YES. but it might and will cause a lot of confusions. Do cat's use the same manner of articulation as humans do?

**Further Reading:**

* Remmers, J.E. and H. Gautier. Neural and mechanical mechanisms of feline purring. Respiration Physiology, v. 16, December 1972: 351- 361.
* Holub, Joan. Why do cats meow? New York, Dial Books for Young Readers, c2001. 46 p. (juvenile).
* Frazer-Sissom, Dawn E., D.A. Rice, and G. Peters. How cats purr. Journal of zoology: proceedings Zoological Society of London, v. 223, January 1991: 67-78..
* De Lanerolle, Nihal C. and Frederick F. Lang. Functional neural pathways for vocalization in the domestic cat. In Physiological control of mammalian vocalization. Edited by John D. Newman. Plenum Press, New York, London, c1988: 21-41
I would like to encourage you in your transcription efforts on cat's phoneme inventory! I am a student of linguistics (B.Sc.) and I have two cats, too - that's why I have been thinking of doing the same one day. We often face men, even scientists, who claim that speech as a unique feature to the human race. When it comes up to explore speech in other species, the research is no longer free or open to new ideas but blocked as the results may question no less than the human supremacy: If we had evidence that various species do have language and use it on purpose (this requires advanced cognitive skills), it would be increasingly hard to keep up the human centric view on the world. So keep on noting down the distinctions that you recongnize in your cat's utterances and do not be discouraged if this would require creating a new kind of IPA chart for the felidae.