i can't believe no one's responded yet. i don't have kids, so was waiting for someone with kids to write. but since no one's stepping up, i'm game - we travel with friends with kids, and also have other friends with kids who cruise royal quite often.
allure has mini golf (good for both ages), a rock climbing wall (for older kid), a surfing simulator (also for older kid), a kids' splash pad and pools (for younger kid), the dreamworks experience (parades and 3D movies and character meets - probably good for both kids), and a carousel (for both again). all of that is included in your cruise fare. there's also an arcade for an extra charge. adults can do all of those things, too - DH and i went on the carousel numerous times.
the only exception is that i think you have to have a kid with you to go in the splash pad/kiddie pool area. allure also used to have kids' cupcake decorating classes the parents could help out with, which i think were about $20 per person, but i'm not sure if they are still going on (there were also adult cupcake classes, which the older kid could probably handle).
the diving show at the aqua theater and the cirque-du-soleil-style show in the main theater are also good for all ages. the aqua theater also has a dreamworks-themed show one afternoon or evening, and has short fountain shows throughout the day and night, and sometimes movies on its big screens. i don't remember if chicago has switched over to mamma mia yet, but i wouldn't recommend chicago for most kids, unless they are pretty advanced (like PG-13 plus).
i believe there are also family events like family scavenger hunts or family karaoke - you can probably find a "cruise compass" (the daily newspaper with list of activities) for your itinerary on cruise critic, or just do a google search for one.