My personal preference for city breaks is for about 3 days, enough to see most of the sights, then go home exhausted! We visited Budapest in winter, so 3 days was plenty. It is a busy city but there is lots top see and it is quite spread out, so going for a week may give you time to see it at a more leisurely pace. I believe there are quite a few thermal baths, some of which have sunbathing areas, so that may solve your sunbathing problem. There are also parks where you can relax and chill out for a while, away from the city and traffic. Going for a week may also give you the chance to venure further afield for day trips.

The Hotel Pannonia has an outside swimming pool, and is situated on the city limits on the Buda side of the river. It takes about 20-30 mins to get into town by bus.

Most city hotels won't have pools, but can I suggest that you investigate the Hotel Gellert which has its own thermal pools attached. You may find a pool at the big chains, eg Hilton. Margaret Island, in the centre of Budapest has thermal pools (outside mostly I believe).

Budapest is probably worth a week's visit, because you can easily explore outside - Szentendre and the Danube bend - or even take 4 days in Budapest and 3 in the Lake Balaton area ... I recommend the Hotel Annabella in Balatonfured if you want to be near the lake shore (private beach) and excellent outdoor pool in a nice garden. (I've been twice but not that recently.)

I did Budapest for a week in August and there was plenty to do. I also did a 4 day Easter break which was about right as the weather wasn't as warm. In August it was 38 degrees for three days and then the weather broke and it went down to 20 degrees. We spent a day in the pools on Margit Island although I wouldn't really bother if its not hot. We spent a couple of days wandering around, one doing the touristy things and one on the danube up to Schezandre. That left a couple for just relaxing - perfect!!

I was in Budapest for 3 days in February. The time was all right, for me it is better to visit cities outside of the summer. In the summer it can get too hot. I could have used one more day for sights and museum, and it is a really nice city. The buildings by the Danube are fantastic! Also the beer and food is cheap bud very good!
For hotels I could suggest priceline.co.uk which has really good prices on 4 or 5 star hotels in the city using the name your own price where you bid for a room without knowing which hotel you end up in.