There are sooo many potential reasons for this lol.

I'm just going to jot down a few here, but I'm fairly certain that I'm missing some fairly important ones:
The main reason I can think of for why he might have chosen Timon and Pumbaa to be Kiara's babysitters is because they're the only friends he can really trust to keep her safe. I guess Zazu's too busy being a majordomo to watch her (he wouldn't be much use as a bodyguard anyway, as seen in the Elephant Graveyard scene from the first movie) and all the other lionesses are either too busy hunting for the pride or don't have the energy to keep up with her. T&P abandoned their jungle home to go live with Simba in the Pridelands, so I figure Simba might have assigned them this task since he didn't exactly want them to sit around and do nothing contributive.

Plus, clumsy and dimwitted as they are, the two proved to be excellent babysitters while he was growing up. They also proved to be loyal and good, when both of them left their comforable "Hakuna Matata" lifestyle to fight alongside Simba and Nala for the kingdom. Alternatively, Simba might have wanted Kiara to adopt the "no worries" approach to life that had kept him happy and safe as a cub, since his own life had been full of such terrible tragedies.
I also don't think any other babysitters would have been as "cub-friendly" as Timon and Pumbaa were. Something tells me that a huge rhino or cheetah guarding the young princess wouldn't have much tolerance for her silly antics. xD Timon and Pumbaa were the best choice since they were both fun, hilarious, and energetic. Furthermore, they were small and lightweight, so there wasn't much of a chance that they might accidently trample Kiara or scratch her. Since they only ate bugs, they were also completely harmless cuddly "toys"

Kiara had no other cubs to play with as a cub, so Simba might of decided that Timon and Pumbaa were a good replacement. He might have even hoped that being influenced by these two at a young age would cause her grow up acting and thinking just like him lol.
Long post is long.
