If you cancel the contract, then you have to pay cancellation charges, its as simple as that. Nobody in BT is going to cancel those charges.
Charges are shown here http://www.productsandservices.bt.com/products/static/terms/contracts-early-termination.html
The only way that BT will allow you to leave is if they agree in writing, that they are unable to provide you with service, or repair your existing service. This is very unlikely as most ISPs, including Sky, use the Openreach copper network.
It would have been better to bring your problems to this forum first, and they would have been sorted.
I've been like this for a while:
I live in a communal block of flats and on August 17th 2016, someone from Kelly Communications came to install a new line or provision on behalf of BT. During the installation, he knocked the wiring where the line comes into my properly from outside. This has caused me to have numerous disconnections and reconnections everyday for the last 4 weeks bearing in mind that I only joined BT on August 15th, coming from Sky.
Since the 17th, I've spoken to about 18-20 BT employees from their faults, technical, complaints and cancellation teams in order to recifiy this. They have already sent an engineer out once which looked at the wiring both inside and out. Didn't find any issues with my internal but the external was suddently corroded. They have arranged for another engineer to come out this Thursday to look at it again and recifiy the issue of the numerous dropouts I'm having. I asked the supervisor to put in his notes on my account to say if this fault isn't recifiied that I will be cancelling my contract, even though I've just paid a £90 bill from them and they issued me a £100 "gift card" as I had signed up to a special deal.
I hope I'm within my rights to cancel without charge as they can't offer a stable service nor the speed they said when I signed up which was 50mb/s. I've had two Smart Hubs from them in the period I've been with them too.
You can only cancel without charge if BT agree in writing that they are unable to provide satisfactory service. You may have to repay any issued gift card
In order to find out what services and speed range you should be getting, please enter your phone number into this form.
Broadband availability checker
Please remember to edit out your phone number before you post the results.
Could you please post the results from the BT Wholesale speed checker, including the further diagnostics.
If your connection speed falls into the impacted range, then more help can be arranged via this forum.
The forum moderators can usually sort these problem out.