I have a
Verilux Full Spectrum Booklight that I ordered online. I like it a lot, although I've only used it once to try it out. You can check out the website for more info; I thought having a full spectrum, or natural, light would be better on my eyes. (It has a blue tint that I didn't expect.)
It is its own battery charger. You can plug it into your computer to charge it if you put in rechargeable batteries. I haven't put rechargeables in yet, so I can't tell you how well it works. It only comes with a cord with a USB cord, so you can't plug it into a regular wall outlet.
One more thing--I have the Amazon basic black cover, and the booklight slides onto it pretty well. With 2 AA batteries, it does make it top heavy, and adjusting the light will still give a glare on the Kindle, but it is high on the screen and if you just tilt the bottom of the Kindle edge away from you, the glare goes off the screen.
I used a coupon code, WB6017, for 15% off my order, so I pretty much got shipping free.
Good luck in finding a new book light!