It's hard to follow your budget as we don't know the number or days you'll stay nor how many people/rooms - is it just you, adult DS, and child DD? Do you need two rooms? Also, it's late to book for this July - you can usually find deals in Malta from November - March, but July is very high season and books up in advance!
Valletta is tricky - you'll likely find more options for a rental flat than for affordable hotels. Try something like booking.com to see what's available for your dates and get an idea of prices. But for cheapest/best value hotels in Valletta with aircon and pool, I'd look at the Osborne (my favourite - they're in a quiet area very close to the city walls for walks and amazing views but also very close to the bus station and the road into downtown Valletta.
For Sliema, if you want a pool but also very affordable prices, try the Carlton Hotel or the Preluna or Palazzo Violetta. Also the Diplomat - but be aware that they are north of the tunnel, so not an easy walk to the main beaches in Sliema.
If you're not finding what you want in Valletta and Sliema, I'd maybe expand your search to all of Malta and see what you find. Gzira is basically an extention of Sliema and perhaps even better located to tour the island - look at 115 The Strand and the other Neu Collective hotels.
You'll usually find the cheapest reasonable (but not too far out geographically) places around St Paul's Bay and Bugibba (still pretty central, just a more built-up area). You could also look at Mellieha, which is quite a bit north and so not as convenient to explore the island, but has some quite nice cheapish hotels.