Taxis in Dubai are cheap. It normally takes aroud 30 mins to get from the airport to the centre of Dubai, but it may be quicker in the middle of the night.
Having just come back from Dubai nat the weekend, I can tell you that most of the major hotels are only about two thirds full, and are discounting their room tarrifs / offering free upgrades etc.
Exchange rates for Sterling to Dirahms are about 1:5, and the rates available at the Dubai Airport or hotels are competitive.
I stayed at the AlQasr, part of the Madinat Jumeirah complex. Has its own beach, and pool both are fantastic. The hotel is top dollar, absolutely fantastic, but can be pricey - £400 per night before any dicounts.
Good luck, and enjoy.