The best restaurant in the area is an Italian- Tentazione in Mill St, opposite La Lanterna. But the request was for Japanese, Thai etc. There is a Thai restaurant on the riverfront by Hay's Galleria called Kwan Thai, which looks a bit business/corporate but served up a lovely meal the one time I went there.
There are a bunch of places on the waterfront--I think Brown's and Ask will be the most kid-friendly. On the inside portion of Butler's Wharf (I think that's still Shad Thames...I forget), there's also a Pizza Express. There is an Indian place (again, name escapes me...Tiger something?) that's decent, but a little formal.
Thai-wise, there's a small and informal place on Tooley Street between Butler's Wharf and London Bridge that I've always liked.
Also, if you keep your back to Brown's and go to the right and cross the little foot bridge, on the other side, make a right. There's a decent local pizza/pasta restaurant. La Lanterna?