OP - your mum has it the wrong way round. If your marriage is going to surivive, it can only do so if he is going to fight for it.
You will get a chorus of MNers telling you you have to end it, and perhaps it won't work out, you need to be prepared to accept it might well be over, the marriage you had is over, there might be a new relationship, but only if he does all the running.
You can not fix this, you shouldn't try. If he wants you, truely, not just staying with you because it's easier, then he'll fight for you.
The marriages that have survived affairs without masses of bitterness I know have involved the person who had the affair fighting for their family. It usually has involved cutting all contact with the OW/M, which in this case would include your H looking for a new job. (and frankly, it's probably the gossip of the office already and made him a bit of a joke even if he doesn't realise, best he starts looking for a new job anyway)
It could well be that while he's panicking about you finding out, when faced with having to choose, he won't choose you if you aren't going to run after him, you need to prepare yourself for that.