I don't think that there's a bike hire shop in Hyde Park; I've never seen one and there's nothing listed here:
http://www.royalparks.gov.uk/parks/hyde_park/s port.cfm
The London Cycling Campaign website lists cycle hire shops in the capital:
http://www.lcc.org.uk/index.asp?PageID=177
Unfortunately, the information doesn't seem to be up to date. I suspect that Banana Rent no longer exists. (The web link is invalid and the phone number goes through to a waterski hire company).
Action Bikes still have a store on Dacre Street (near New Scotland Yard) but their website doesn't mention bike hire. (It might be worth a try though).
Go Pedal don't seem to have a base open to the public. Instead, they offer a 'deliver and collect' service. That might explain the rather hefty price of �24 for one day's hire. (Interestingly, their website's homepage actually shows a map of Hyde Park).
The London Bicycle Tour Company is cheaper, at �16 per day, but their base is south of the Thames, near to Waterloo station.
On Your Bike's website doesn't mention hire services but they could be worth calling. Once again, though, they're on the south side of the Thames, near to London Bridge.
Your best option might be to consider hiring a folding bike from Velorution. They're based near Oxford Circus and charge �20 per day (and not the out-of-date figure shown on the LCC website).
Chris