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".
The hiss sounds like an [x].

The best purring sound I can make is the Spanish trill [r] with my mouth almost closed, or modulating my mouth so that it fills with air, it leaks out, repeat.