If your after a central London experience outside of central London, you're going to be disappointed.
Birmingham hadn't got that much stuff to keep you occupied for that long. You definitely couldn't spend endless weekends in the city centre. It's tired, run down and nowhere near the city it used to be. The whole centre used to be full of shops and independents and markets and now it's just grim. Endless pound shops, phone shops, a big hotel in the centre houses immigrants waiting to be processed, streets stink of piss, homeless and beggars everywhere.
Bullring takes about an hour to walk round and nothing exceptional in there
Custard Factory worth a look at, maybe takes half a day? Museum and Art Gallery is currently closed down. HS2 works are currently dominating and making it look even worse
There are a few little Museums but the problem is they never change inside.
When the bmag closed down, the displays in there had been the same since the 90s and were broken and tired.
You would be hard pushed to find something every weekend for longer than 3 months and that's being generous.