There are no kid-friendly ereaders. That said, preschool children can learn to be careful with them--as long as the first several times they're used are under direct guidance, and the next several have supervision, and you're braced for being willing to replace it anyway in case of an accident. ("Walking through house; bumped shin; dropped ereader; landed on ereader.")
I believe the Kindle Fire, which is a multipurpose tablet much more than a dedicated ereader, would be best for that. However, if you really can't afford to replace it, don't share it with the kids yet. There's no guarantee you'll have to, but it only takes one clumsy moment to kill the device, and with three kids under five, you've got about a 10-year span of clumsy moments ahead of you.
Ereaders aren't incompatible with a rough-and-tumble kid-friendly household, but the phase of "get kids used to the fact that this device does not go in the dishwasher" could cost you the first one.
Option 2: Buy a refurbished, older model from Amazon (like a
used PRS-300) and see if that works; find out what you like & don't like about it from that while you're shopping for a newer model with more features.