I agree with Gary on this one.
You must price a job by the amount YOU need to survive in business, NOT by what you think the client is prepared to pay. If they can't, or won't pay the price you need, then don't take the contract. There are others that will be prepared to pay you a fair price. Just keep looking and forget about what you 'think' the going rate is.
A couple of tips:
1. Try starting out by approaching new businesses into your area, not existing ones that already have cleaners.
2. Price by the job, not by the hour. You still estimate your price by the time you expect it to take, but there is no need to tell the client this information.
3. Draw up a list of tasks and provide this with your quote. The client then understands that this is what he is paying for, not the time it takes to do. Using that basic list, you can then negotiate items, in or out, with the client and adjust your quote accordingly.
4. £15 per 'estimated' cleaner hour would not be an unreasonable price to aim for IMO. You will NEED to achieve somewhere near this amount if you intend to (legally) employ cleaners to continue this contract in the future, when you are unable to do it yourself.