All of the above posts have recommended expensive LED lighting fixtures that cost about $300 to $400. While these are good fixtures, it's really a waste of money to put such a fixture on a 10 gal tank.
You don't mention exactly what kind of sea anemone you have. If you have a typical clown hosting anemone, such as a BTA, it's going to out grow the tank. You would also have the problem of maintaining water quality for an anemone in a 10 gal tank.
If you have anemones such as rock anemones or mushrooms your fine.
As for "all corals (that grow) in a 10 gallon aquarium", your system is going to be best suited to soft corals, and LPS corals, also related livestock like zoas, mushrooms and polyps. SPS corals are a bit of a stretch, mostly due to water quality issues.
Considering what you can reasonable keep in the tank, and that you want something reasonably priced, go with something like a 2 bulb T5 fixture. Here is one example (offsite) -
http://www.marinedepot.com/Coralife..._Fixtures-Coralife-ES08600-FILTFIT54U-vi.html
There are plenty of others. Note that this is a 24 inch long fixture, and it will overlap the tanks, but since the legs are adjustable it will work just fine.
Keep in mind that normally I'd not recommend a 2 bulb T5 fixture, but a 10 gal tank is only 12 inches deep an 10 inches front to back. You'll get plenty of light, and at a price a lot less than the LED fixtures recommended.
Noe if you got to have LEDs, Then consider a one or two
Finnex Ray2 fixtures at about $75 each. Here is an example (offsite) - http://www.amazon.com/dp/B00845V426/?tag=reefsanc-20
Granted the Finex Ray 2 is not going to be as nice as some of the other recommended fixtures, but they are a comparative bargain, especially for a 10 gal tank.
In the future, if your looking to upgrade to a much larger tank, we can talk about all those expensive LED lighting systems.