It is possible to replace the hinge, but it is a couple hours of tricky work. If you are not comfortable completely dismantling your computer, I would take it to a repair shop. You can usually find the hinge on ebay. Just search for "thinkpad x230 tablet swivel hinge part".
I replaced the hinge on my x61 tablet without any trouble, but I am very comfortable with repairing computers. Take lots of pictures to make sure you get it back together correctly, and keep track of where the screws go, there are a lot of them. Good luck!
Here is the repair manual for the x230:
https://download.lenovo.com/pccbbs/mobiles_pdf/x230t_x230it_hmm_en_0b48730_02.pdf
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