...or Addison Lee offer a superb service for this with trusted drivers.
Are you a job agency?
The trouble is that you require a driver who supplies their own car and maintains it and is working for hire and reward therefore must be licensed to TFL and more importantly insured to carry fee paying passengers,
Also they are required to turn up ad-hoc for anywhere from 1 to 10 hours of employment day or night.
As an example licensed chauffeur companies charge a minimum of £400 a day for this service but that is only for the hours of 9am to 5pm...costs increase after 5pm and increase again after midnight (admittedly yo get a Mercedes as your chauffeured car).
If you want to keep within the law I suggest a cheaper taxi firm.