It is likely, to be honest, that the contractor is trying to fob you off with a bodge job to fix an error on his part, and you shouldn’t stand for it.
What is a Soakaway?
A soakaway is basically a trench filled with a semi-permeable substance or structure which can collect excess water and slow it down on its way to the water table. Many people think it’s a pit with some gravel and, for gardens with a mild Drainage Problems, that can work. But a modern soakaway is more likely to be an open chamber with holes or slots at strategic intervals in the sides.
The contractor should put a proper soakaway in an investigation pit dug to find out how far down the water table is, to make sure it is a variable option. Assuming it is, then calculations taking the area that needs draining (perhaps not the whole of the driveway, depending on the Fall Of The Drive) and the average rate of rain fall in storm conditions (generally a figure or 5cm per hour in the UK) will give the necessary volume of the soakaway.
Given that your contractor had apparently not done the correct calculations when setting the levels on the driveway (or you wouldn’t be in this mess), it seems unlikely that he will be approaching this soakaway solution professionally either.
You are probably aware that the correct thing to do is to ask the contractor to relay the drive properly, or refund you so that you can get someone else in to do it. Go to the Citizens Advice Bureau, where you can get help to set out your arguments and get what’s rightly yours.
If you don’t feel able to do this, then a better solution than a soakaway might be to take up the concrete that’s been put down close to the garage and re-lay it with a drainage channel (or more than one) to a run-off on one or both sides of the garage.
Not being able to see the site makes it hard to know if this would work, though. If it would, and the drive is new, then the contractor ought to be able to relay the re-contoured section without it showing as a patch. He will need to be careful to take the concrete up at the joins in the pattern then match the new again, and how successful this is will depend on the pattern, but also the level of skill and amount of preparation put into it.
Past experience makes it improbable that the appropriate levels of skill and preparation will be deployed though.