Hi, I'd like to help you with your question.
It sounds like you are a very good friend to this person, but she is unwilling to get help for herself. You are right, there is no way to force her. However, there are some things you can do to try to help her.
One, it sounds like she may abuse alcohol and this is probably the source of her mood swings. Often, alcohol use can cause a severe change in personality. A person in a great mood will suddenly start to cry, or someone who is usually easy going will become angry and destructive. When your friend uses alcohol, try not to engage her in conversation. See that she is safe and comfortable then let her know you will see her again when she is sober.
You can also suggest that she try to become sober by attending AA or seeing a therapist. However, given that she has refused so far, she may not be willing. But suggesting therapy can at least put the idea in her mind.
Since alcohol is a depressant, it will be difficult to tell if she does have a problem with depression at least until she can either be seen by a therapist or stop using long enough for the alcohol to be out of her system.
You may want to contact Al Non for some ideas on how to help your friend. The link is at http://www.al-anon.alateen.org/.
Continue to be there for your friend and talk with her about therapy. You can help her find a therapist by either having her talk to her doctor about a referral or if she attends church, her pastor can help. Or on line at www.find-a-therapist.com. If you can help her find a therapist, she may be more willing to go. Seeing a name and calling for an appointment may make it more real to her. Also, offer to go with her for support.
Don't forget to care for yourself while caring for your friend. It is easy to become burned out when your relationship involves such intense moments. Take some time to pamper yourself. Maintain your other friendships and relationships. If you find yourself becoming upset and angry dealing with your friend, take a break from your contact. You need to be sure you care for yourself so you can be there for your friend.
I hope this helped you,