This one's a quickie: If the singular person I address in an essay (one, one's in this instance) is a he in one paragraph, can he be a she in the next? I'm wondering if changing it up that much is frowned upon as confusing or simply doomed as an inconsistency like using you and one? Fur random bad-zamples: They aren't only good for cuddling, though: one's cat can also be his saviour. Housecats have saved numerous lives in the past, so one would be smart to buy her cat lots of treats and change its kitty litter often; it is always a good idea to stay on a cat's good side.
Yeah, I don't like the mixing, especially the combo of his and her. It feels as though one is going very out of one's way to be politically correct with gender representation. It detracts from the essay itself. You might consider rewording the sentences so that they don't require the pronoun being said twice. A cat can also be one's savior. ...so it would be smart to buy one's cat lots of treats and change the kitty litter often; it is always a good idea to stay on a cat's good side.
I usually do what Wrey did in his very first example (use "his") when writing an essay or something formal. On this forum, I use "his/her" a lot, but truth be told, I do this mostly out of political correctness than anything else. You can also rephrase it like Wrey did in his second example. In fact, in this case, his second example is the one I would go with.
I use 'his', since I'm a him. I figure if everyone does the same (men always use 'him' and women always use 'her') it'll just about average out in overall usage. I also like @Wreybies' solution, although the 'one's' can sound a little stilted, depending on context.