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: 
If you'd like to transcribe how we perceive cat language (as opposed to how cats do), you might be interested in the corresponding problem for bird song, which has at least some literature. I found a reference to an LSA talk from 1977 by Donegan and Stampe, "Old Sam Peabody, Peabody, Peabody".
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.