Every spiritual gift has its own difficulties and temptations. If you have the gift of mercy, then Paul's words here are probably for you:
6 And we have different gifts according to the grace given to us. If the gift is prophecy, that individual must use it in proportion to his faith. 7 If it is service, he must serve; if it is teaching, he must teach; 8 if it is exhortation, he must exhort; if it is contributing, he must do so with sincerity; if it is leadership, he must do so with diligence; if it is showing mercy, he must do so with cheerfulness (Romans 12:6-8). 14 Do everything without grumbling or arguing, 15 so that you may be blameless and pure, children of God without blemish though you live in a crooked and perverse society, in which you shine as lights in the world 16 by holding on to the word of life so that on the day of Christ I will have a reason to boast that I did not run in vain nor labor in vain (Philippians 2:14-16).
Servanthood does mean putting the interests of others ahead of your own (Philippians 2:3-4). But that means risking rejection when you see someone doing something that is wrong, something this is bad for them. For example, confronting an alcoholic is more loving than leaving them alone, to destroy their lives and others.
The loving thing to do is to sit down with this woman and explain how she is selfish and demanding, and how this affects your life and hers. No doubt this woman keeps wearing friends out, and moving to others, wondering what happened. You can help her end the cycle. If she is not a Christian, this is an opportunity to show her how sin is destructive, to her and to others. She may need to trust Christ, as well as to mature.
I'm not sure that I really like the word "boundaries," but I'll use it for the moment. There probably need to be some boundaries established. You need to make it clear what you can and will do, and where your limits are. There are some things you simply cannot do. You can't drive her around when you are at work, or at school. You can't do things that cost money you don't have. You should not do things that are teaching this woman bad attitudes and actions (like being selfish).
It sounds to me like she needs a friend honest enough to confront her in those areas where she "uses" other people. She, too, needs to learn to be a servant -- assuming that she has trusted Christ and is a true believer.
You may very well find that this woman has already worn out her welcome with others in the church. Perhaps another friend from church can help you sit down with this woman and truly minister to her. Why not have a Bible study with this woman? You might start with a Gospel (like John), if she is not saved. Or you might go to a passage like Ephesians 4:17 -- 5:21, which describe the way our new life in Christ should be different from the way we once lived as a non-Christian.
Here are some verses about correction to keep in mind:
I know it is much easier for me to tell you these things, than it is for you to do them. Nevertheless, this is what being a Christian friend is all about. Your friend may thank you for telling her what others would not. (And, she might get angry with you.) Either way, you have done what our Lord told you to do.