I've been away a couple of days, and the tick-parts-left-in problem may have resolved by now. But I wouldn't try too hard to get the tick's jaws/head out. The jaws have little barbs in them, and digging them out probably causes more trauma (psychic as well as physical) than it's worth. Just clean the area with soap and water (or Betadine if you have it, it's sold over the counter in pharmacies) and keep an eye on the area for infection, which is unlikely. What usually happens is that the dog's (or human's) body reacts a bit, and a lumpy scar may develop, which takes a long time to resolve, but this is not dangerous. The bad stuff in ticks is in the mid-gut, not the jaws or head ==which is why it's disrecommended to squeeze the tick when removing it. As several have noted previously, the easiest tick-removal tool is a Tire-Tic, sold in most pharmacies--be sure to get the prongs of this as close to the skin as possible, you can usually remove the whole tick this way. The instructions say to twist the device (? counter-clockwise?) when pulling, and some have said this works better, but I usually just pull firmly but slowly without twisting, and very rarely do I find that the tick breaks.
Garlic does contain some stuff which is toxic to cats, and to a lesser extent, dogs. You can google it to see.