You have known about your former husband's living arrangements for over two years. It is possible but not probable that "in the beginning" the relationship between him and this woman was platonic. More likely, after 15 years of marriage, he didn't want to drop the bomb on you that in addition to not wanting to live with you any longer, he also had a girlfriend.
I'm sorry that this is not a comforting thought, but if my man left and moved in with another woman over two years ago, I would have smelled a RAT! The RAT is your former husband. The LIAR is your former husband.
If your former husband is a good father to his children and his girlfriend poses no threat to their safety, I would suggest that you be civil to the woman. There is no need to form a "friendship" with someone with whom the only thing in common you have, is your former husband.
Your children love you and their father. They have seen your pain and anger and don't want to be disloyal. If you say and do things to turn them against their father's girlfriend (who may become his wife), it will only cause them additional pain and to be torn between mom and dad.
Don't live a life of bitterness, start building a new life of your own. If you do not wish to date someone else, get involved with other things that will get out of the house and socializing. Be good to yourself.
Blessings....