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You married your spouse, not your family. While his or her family members are part of the package, they are not a part of your intimacy and they do not share the same journey with the two of you. If you make it very obvious that you're not bothered by jealousy, insinuations, rumors or gossip, it will soon become clear to your S-I-L that her barbs, attitude and meanness aren't pricking you in the way that they used to. Eventually, it ceases to be profitable or enjoyable for her to keep bothering and most likely she'll begrudgingly go and find someone else to taunt and hassle.  Spend less time around your S-I-L. In what ways are you putting yourself in her pathway? While it may feel like you have to put up with her, you can find ways to reduce the time spent together. For example, ask other family members to meet you at different times than when she is around, more often than not. Don't always do this, or she will have a legitimate cause for complaining, but time spent with other family members shouldn't always involve her presence. If you live far away and have to visit once a year, stay in your own accommodation to give yourself respite. Take walks, get outside and don't overstay any welcome when it comes to drawn-out family events that press your buttons. Families know the pressure points better than anyone and unfortunately, some like to press them. At such events, your S-I-L probably has alliances that she can set in train to be even more effective, so the less time spent near such complaint-prone cliques, the better. When you are around your S-I-L, try active listening and acknowledgment in place of letting your fog of self-defensiveness take control. When she gets on top of her complaining mountain, instead of trying to topple her off with "if you think that's bad, you should live in my shoes" replies, actually focus on her and try to discern what is really driving her jibes, whining and gossip. By not making this about you, you may be truly surprised at what you unearth.As for responding to her, acknowledge her pain with neutral comments like: "I'm sorry you have had to go through that to pay an electricity bill. It must be hard having four kids chewing through the power each month." Don't offer advice, don't offer how you would deal with it and don't ever offer to pay or pave the way to see her problem resolved. She owns it, you simply acknowledge it. If your S-I-L has been a pain more than once and has even done things to show you up or drag you down, the chances are that she will try to do it again, even when you don't bite. But if you're ready for it and if you're understanding as to where she is coming from (insecurity, loneliness, feeling left out, needing to be in control, etc.), you can be compassionate about her actions and detach yourself from her drama. If you don't carry her load, she'll be forced to do it for herself and will stop seeing you as a viable target.
Get on with your lives together. Listen for real. Be compassionate.