Think of it as a good motivation to learn the joys of manual focus. Since the widest aperture is not very wide, manual focus will not be as critical as a f/2.8 lens. I know a lot of hobbyists would denigrate the 55-200 because it was an inexpensive "kit lens" it actually is a pretty decent lens, and in its range, better than many of the more expensive wider range lenses that go much wider at the wide end. Newer designs of lenses use a Silent Wave motor for focusing but yours used a dc motor that couples to a ring gear. There are 3 versions of that lens, the original non-VR, the VR and VR II. If yours was the kit lens included with your camera it is probably the VR version or original if the focus pinion gear broke since the later VRII has manual focus over-ride which means it is not damaged if attempting to manual focus while the focusing motor in operating.. The main differences include lack of noise, and quickness of driving the internal lenses and the Silent Wave can be manually focused by just grabbing the focus ring and turning it, and internal motor can be easily over-ridden without breaking the teeth of the pinion gear as has happened to your lens.
Repairing that lens is going to be more expensive than its value which is not such a bad thing, it means a used 55-200 VR II can be purchased for less than repairing yours. You can find a used replacement for $60-80 on eBay or for the newer VR II version for around $100-110. Some are listed for bid starting at $15. A higher performance lens in that range will be a lot more expensive and the only advantage most would see is being more effective in low light. For casual users, the 55-200 VR or VR II is an ideal compact light lens that is a very good compromise of performance and size/weight. Any other lens covering 200mm will be larger and heavier. That lens is capable of very good portraits in the 80-150 mm range, and with pretty good background blur if you get your subject closer to the camera than they are to the background.
Good luck with your new---to be---lens.
If your favorite photo subjects are something specific such as flying birds or action sports, the autofocus speed would be better on higher-end lenses, but higher end means expensive. 10 times as expensive as a minimum. If you are interested in stepping up, the 70-200 VR II f/4.0 is about $1100 and the more popular and heavier 70-200 f/2.8 is north of $2400