Hi Roo,
Firstly, you do not need to knock the wall.
How long since you noticed the problem?
Any chance that the gutter is broken and water is coming down the wall to cause the problem?
What sort of condition is the gar n wall? Solid/Crumbling etc.
What is condition of rear wall of house?
What materials were used in garden wall and rear wall?
It would be unusual for the garden wall to be tied into the back wall. They rarely did in the old days!
It should be possible to rake out the pointing between the wall and the house and fill with a waterproof or tanking cement which should stop your problem.
It may also be sufficient(temporarily) to just repoint with silicone the top and side of the wall where it butts the house.
If raking out the joint between wall and house, it may be a good idea to cover with plastic and leave for a while to dry.
In my experience, the materials are expensive but the labour from a specialist contractor is extraordinarily expensive.
There are a number of companies out there. I have no affiliation to any but I can happily provide a recommendation, if you wish.
Best of luck.