Not only should you throw him out to fend for himself, you should pursue him for child support. Maybe if he'd been working as much as you have, he wouldn't have time to pursue a woman who is not only emotionally unfaithful to her boyfriend, but is also willing to stay involved with a married man.
Take some time to consider exactly why you still love this guy. He is using you for emotional and financial support while flaunting his online relationship with another woman. By your description of him, he is not worthy of you being his wife or best friend.
The longer you hold onto him, the harder it will be to move on with your life. Find friends and/or a support group/therapy. I hope things work out well for you and your kids.