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If the needle has broken off and is still in the leg, then it needs to be removed. Since it was sterile, it will not cause an immediate problem, but over time, any foreign body may serve as a site for infection, and the sharp points of the needle can cause trauma or bleeding over time.
Since you know the location and direction of the needle insertion, it would be fine to try to remove it yourself. You can try to pressing down the skin and subcutaneous tissue around the spot of insertion to see if it can be grabbed by some tweezers. However, if you cannot get the needle out, then it would be better for you to be seen. Since you are not having pain, it is not necessary to be seen in an emergency room in the middle of the night, but you should contact your doctor first thing in the morning. If your doctor also cannot feel or see the needle on exam, then an x-ray can be done to determine the exact site of the needle.
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