Hello!
I'm from KH, and live in Moseley now and I love it here! But I get that it's not for everyone.
In birmingham you could look at Harborne (close to edgbaston and the city centre, lovely houses and great high street, but pricey) and Bournville village trust properties.
Edgbaston is dodgy: even the nicest houses are on the edge of very inner city housing, the busiest roads and it is generally quite rough IYSWIM?
Out of birminham look at Shirley, between Brum and Solihull, good high street, fantastic park, every supermarket within reach, great schools etc.
There's also Dickens heath, not my cup of tea, but it is popular for a reason.
The other side of town there is Bearwood. Beautiful Victorian houses, Worley woods, close to city centre, a reasonable high street and cheaper.
Finally, you could look at hollywood/wythal. More village feeling, but too suburban for me. Close to the motorways though, good schools and close enough to the city centre to not make it a ball ache.
Let me know if you need more! Best of luck!