It's happened to almost everyone- while petting your dog you find a tick and after the initial "ick" factor, you pull it out- but the tick splits and part is left embedded in the dog's skin. No matter how carefully or by what method of tick removal you use, it may be possible to leave part or all of the head behind- especially when removing it from a squirming dog! Some veterinarians might recommend taking the dog immediately to a veterinarian for tick removal- but letting time take its course is probably the best option for a healthy dog.
1. If you've just had a failed attempt to cleanly remove a tick, don't panic! Just clean the skin around the bite with rubbing alcohol (on a cotton swab) or hand sanitizer gel to reduce the risk of infection.
2. A dab of triple-antibiotic ointment (the human kind is safe for dogs) to prevent infection where the skin is broken can be added twice a day until the wound closes. Monitor for the development of a rash which can signal infection with lyme disease. (But don't panic, a tick must be attached for 48 hours, whole, to begin transmitting lyme disease, and most estimates are that only about 10% of dogs exposed to the bacteria will get sick.)
3. As it heals, your dog's body should form a hard layer of skin around the bite. The hard lump is the body's way of quarantining the tick's head as a foreign object. As layers of skin die and regenerate, the lump will eventually just fall off. This will occur after several weeks or months. If you want to speed the healing process you can break open a capsule of Vitamin E and apply dab to the site once a day. If at any time the bite becomes inflamed or appears to be infected you should consult your vet immediately.
From a vet Q&A at: http://www.vetinfo.com/lyme-disease-rash-in-dogs.html: "The bacteria can be transmitted to the dog only if the tick stays on the dog's coat for at least 48 hours. Bacteria are transmitted when the tick sucks the dog's blood. Only 10 percent of dogs exposed to the bacteria contract Lyme disease.