Problem is: your children probably don't get a word of italian. So you're screwed. And here comes my valuable advice. Take your car, cross the border (10 minutes and you're there) and set "Planet TUS Koper" on your GPS. It is the nearest (and pretty nice) Slovenian shopping mall. Its cinema is called Cineplexx and you get (some) movies in english. Wow. Now the problem is figuring out (on their slovenian website) which are actually in english and which are not. Their number is +386 (05) 973 44 00. Give it a try.
Oh, almost forgot, in the "Triestevillas tourist services" section, you'll find the babysitter option.