I used to live in Las Vegas and while living there my daughter was 1-3 years old. There are a ton of things to do with small children.
The Rainforest Cafe in MGM is an amazing place to eat with small children. It's totally kid friendly, and there are tons of animals (fake) that move around, and thunder and lightening storms happen during your meal. It's really cool, but the line to get in is usually crazy, so make reservations or get there early! There's a real lion at the MGM as well.
The gardens in the Bellagio are beautiful and my daughter used to love looking at all the flowers. I believe this is where the bird exhibit is too (you walk through and there are all kinds of different birds flying around, swimming)
The fashion show mall has live fashion shows (they have a website that you can find the times of the shows), with loud music and flashing strobe lights that kids usually like looking at. And the mall is super nice anyway.
There are free shows everywhere on the strip that just happen out in the open for all people to see. There's a show about the roman gods in Caesars Palace at the end of their shopping center that has lots of fire (might be a little bit scary though).
There are shows in the Venetian's shopping area in St. Marks Square with live singers, people walking on stilts, jugglers, dancers (I used to perform in this show playing flute :-). There are also "statues" in the Venetian shopping center - real people that just stand there for about 4-5 hours straight looking like statues...
Several of the hotels also have drop-in child care centers (many 24 hours) if you want a little adult time. :-)
There's also the Treasure Island Pirate show, and the Bellagio water show (both outside).
If you want to go off the strip, there's the Freemont Street Experience which is really cool... they also have live music downtown all the time. There are also bowling alleys in a few of the casinos (Green Valley Casino in Henderson and the casino in Summerlin)... but these ones are off the strip. I'm not sure where you're staying...
In New York New York there's a HUGE arcade.
The Knights of Excalibur show is awesome for kids... knights on horses in a jousting show and a really good meal is included in the price of your ticket. This is at the Excalibur.
Let me know if you need more suggestions!