I have a lot of sexless characters, as you've probably noticed, and I've come up with a set of pronouns I like for them (although I'm not quite consistent about using them yet). There were two goals I had in mind.

One was to make them resemble the word "it", which is already a third-person singular pronoun - yes, I know about singular "they", but it doesn't really sound right to use for a character whose gender is known, just known to be null. The other was to avoid homophones of existing words, which is why "shi" and "hir" and so on were out. The result is the set ie, ir, and irs, pronounced like she/her/hers but with the initial consonant dropped (or, if you like, replaced with a glottal stop if it follows a vowel, though that's not an official rule so much as something I end up doing when I say them out loud).