There is no perfect way out of your dilemma. First, you should write “Neither Jake nor Janet...”. Various ways through have been tried. Of these, “... have brought their homework ...” is the most popular, clumsy though it still seems. Your ‘his or her might scrape through, despite the danger of a perceived uncertainty about the gender of one or both. It would be great if we could abandon gendered pronouns altogether, as the much needed pursuit of gender equality has complicated our our use of pronouns in many other ways also. “... its bone...” would have been fine if Jake and Janet had been dogs: why not humans too?
Well, we are not quite ready for that. So “his or her’ will do, or, at a stretch, “...his/her...”, but “...their...” is the one with social momentum, I think.