First I should say that anything I suggest is old knowledge, so please do check that these places are still going...
Nothing special, but as most kids will eat pasta, might be worth knowing that there is a Spaghetti House in the area. It used to be on Jermyn Street (maybe still is), but I have a vague recollection that that is also one on either lower Regent or Haymarket. The concierge at the hotel can tell you.
There are also, of course, Pizza Express branches everywhere. I believe there's one just up from Trafalgar Square on Wardour Street (before Shaftesbury).
Not sure how adventurous in eating your kids are, but there is a Wagamama on Lexington Street...
You could head down Picadilly (away from Picadilly Circus), and go to the Hard Rock Cafe...not been there in a long time, but they used to do good burgers, etc.
On the south side of Picadilly (sort of across from the Royal Academy), there was a place called Richoux, which I used to go to for breakfasts especially. No idea how their other food is.
Also, on that side of the street is a church (can't remember the name, but it's the only church on Picadilly), and it used to do a good lunch (soups, sandwiches, quiches, etc) for quite cheap.
I think Fortnum's restaurants are now closed for refurb, but if not, it's worth checking out as they had several good restaurants that might be OK for kids.
Go into the big bookstore on Picadilly (not the famous old one, but the newer one, the name of which escapes me), as I think they have a nice place to eat on the top floor.
That's all I can think of at the moment....but, have fun!