Several excellent, inexpensive hotel swimming pools are available to all visitors to Luxor - not just hotel guests. You can expect the use of swimming pool and Jacuzzi to be about £1.50 / day (GBP).
Private holiday apartments also may have a private swimming pool for the use of guests.
However, swimming in the Nile is definately not an option!
The river has a strong current, dangerous traffic, and slower-flowing parts may not be healthy for swimming - although you will see Luxor people swimming in it.
Instead, use one of the very good swimming pools in Luxor.
To combine swimming with sightseeing, spend a day at the palm-fringed 'infinity' swimming pool at Crocodile Island. At around £10 /day (GBP) it's more expensive than the swimming pools on Luxor's West Bank, but does give you access to the mini-zoo and all the other tourist facilities as well as the swimming pool.
UPDATE: Crocodile Island has been re-named 'Kings Island' by the new owner of the Jolie Ville Hotel, Crocodile Tourist Projects Company (delisted by the Egyptian Exchange from 5 Feb 2010; HKS-Group - a Private Sector Egyptian investment Group). We don't know how peaceful the island will be in 2010 - in January bulldozers and earth movers began work on a development which, over the next 2 - 3 years, will double the number of residential units together with re-landscaping of the entire island. We are also concerned about possible damage to the birdwatching habitat.
Swimming Pool on Crocodile Island ( 'Kings Island' )
Approaching the island