It's fairly common for a tick's head to remain embedded in your skin when you pull the tick itself away with tweezers, no matter how careful you were about the extraction attempt. If you inspect the tick bite and see that the tick's head or mouth parts are still embedded, don't panic. You're just a few simple steps away from removing the tick's head from your skin.
Sterilise the needle by holding it close beside--but not actually inside--a candle, lighter or stove flame. Alternatively, you can soak the pointed end of the needle in a small cup of rubbing alcohol for about a minute.
Use the sterilised tip of the needle to carefully fish the tick head away from your skin. Insert the needle into the first or second layers of your skin just beside the tick's head, no through the head. Think of it as removing a splinter, and try to get the needle "hooked" under the head or mouth parts to pull them out.
Repeat the process, including sterilisation, if necessary, with fine-tipped tweezers; some people will find one method or the other to be much easier.
Wash the bite area and your hands thoroughly with soap and water once the tick's head has been extracted.