The service is on the line for him to use, and has been from the beginning of the contract.
He can use any ADSL hardware he likes - BT Retail use genetic user names and passwords.
As I said in my earlier reply, mistake does not entitle a party to contractual remedies. The law recognises that some people are incapable of entering into a binding contract as they lack the mental capacity to do so, but the Mental Capacity Act 2005 standard for incapacity is high and a contract is only voidable if the counterparty should have known the person lacked capacity (or the person's affairs were under the Court of Protection).
I don't think we're talking here about an MCA 2005 situation - just someone who was maybe a little naive and who failed to check their contractual paperwork and bills carefully. Unless I've misunderstood the scenario and the MCA 2005 potentially applies, the person had ample opportunity to realise a long time ago that they were paying for broadband and could have contacted BT Retail about the matter. They did not do so, and BT Retail continued to incur wholesale costs in providing the broadband service.
The law on distance selling provides the remedy for misunderstanding or mistake when entering into a contract for service at a distance, though the right to cancel under these regulations can be curtailed when the provider has taken steps to arrange provision (in which case they should draw your attention to the limited right to cancel).
The OP's friend could seek a gesture of goodwill, but I would expect BT Retail to resist this. Unless the MCA 2005 applies, BT Retail have every right to enforce the contract, they have incurred costs in providing the service and they have no obligation to protect consumers from their own voluntary acts.
If the OP's friend has been stupid and naive, he should, in my view, take this as a relatively cheap and low consequence lesson that contractual commitments are binding and rights to cancel disappear unless you take timely action. Making a mistake in connection with something like a financial product could be immensely costly - if you discover something wrong with a motor insurance policy when something has gone wrong, it could cost you a £200 fine, 6 points, significantly more expensive insurance in the future, the loss of the premium and personal liability for any losses that would have been covered by the policy if all the details were correct.