Pretty much any time a contractual, statutory or case law precedent uses the word "reasonable" what it means is that the decision regarding what is and is not reasonable is vested in the trier of fact (i.e. the judge in a U.K. landlord-tenant case) to decide on a case by case basis in light of all of the facts and circumstances presented at trial, rather something to be decided as a matter of law by an appellate court absent of extreme abuse of discretion by the judge in determining what is reasonable.
Put yourself in the judge's shoes and consider the range of opinion that people who are judges in a case like this might have, and you have your answer.
A reasonable amount of time balances out factors like the availability of repair contractors at a reasonable price, the magnitude of the repair, and the extent to which the repair impairs the habitability of the premises, and assumes a time frame that would apply if the landlord is diligent and trying to solve the problem in good faith. A reasonable time to fix a broken heating element is different in July than in January.
If three different inspectors failed to determine the cause of the problem, that is going to work in favor of the landlord for a longer period of time. So will the significant extent of the damage (apparently). On the other hand, the fact that the repair rendered the premises completely uninhabitable argues for a shorter time period.
Starting late to address an urgent problem could be breach, although it doesn't sound like this has happened here if some action has been taken that would seem to be reasonable under the circumstances and the results have been inconclusive about what to do. Unnecessary foot dragging could also be a fact in reasonableness, because ultimately the duty is to get the job done.
There may also be a parallel duty of the landlord to provide a habitable premises and to compensate you for time when this is not provided, in addition to the duty to repair within a reasonable time. Asking to be paid for alternative accommodations while diagnosis and repairs are underway would be a logical demand to make before suing for a fairly small dollar amount in an ongoing lease.
You would probably focus on failure to repair in a reasonable time more if you seek to terminate the lease entirely and rent elsewhere instead in the face of repair delays, hoping to avoid the duty to pay rent for the remainder of the lease term due to the landlord's breach of the duty to repair under the lease.
