Don't withhold rent, this is illegal.
What you must do is put in writing to the landlord that unless the problem is fixed within a reasonable time that you will get 2 or 3 estimates to repair the problem and get the plumbing fixed yourself. State that you will deduct the cost from the rent. If she as already had 3 weeks and done nothing then I would give perhaps 3 or 4 days to rectify the problem in your letter. Obviously you could have done this sooner if you had known. This is the legal way of approaching the problem.
You can get evicted for not paying the rent, so dont go down that route. It would take several months and cant happen within the fixed term of a tenancy but could still end with eviction. Use the proper legal process.
Also, you wont be 'turned out' while it's repaired. In extreme circumstances, if the property becomes uninhabitable, then the landlord is obliged to offer you alternative accommodation. Also, tenants are not responsible for fair wear and tear, only damage. I do wonder sometimes about some of the 'advice' that's given on here.
As this landlord sounds a bit irresponsible, are you certain that your deposit is held in one of the 3 Schemes and that you have the legally required Gas Safety Certificate?