Should I make my partner go to counseling?
If your partner says they want to stay in your marriage/relationship, you should ask them to attend counseling but only insist on 2 or 3 sessions. After that, let them know it's up to them whether they continue, and then be careful not to guilt them into continuing if they choose against it. Continued counseling with someone who doesn't want to be there is fruitless—a waste of time and money.
Asking them to come to a few initial sessions is reasonable because they may discover it's not as bad as they thought it would be. Maybe they'll even find a measure of hope that things can change.
Here's what you can do to encourage a more favorable reaction to counseling:
Invite them to be part of the process of choosing a counselor.
Choose a counselor who is experienced and is known for making people feel comfortable in therapy.
Clearly state that you are only asking your partner to commit to a few sessions, and then they are free to decide whether or not they will continue.
Make sure you don't use counseling as your chance to criticize your partner constantly. As much as you might want to vent in those early sessions, be willing to listen and learn. A good counselor will guide the process.
If your partner refuses counseling, ask them to devise a plan for both of you. Make it clear that you require more than "let's just forget what happened and move on." Alternatives to counseling include support groups, couples retreats, books (see my recommended affair recovery books), or mentoring by a couple in a healthy relationship.
If your partner simply refuses to take the lead (or, at least, actively cooperate) in the healing process, you should question the future stability of your relationship. In my experience, the best indicator of successful affair healing is the willingness of the involved spouse to actively participate in activities that lead to positive change.