To answer the question in your comment to Jessica, it is a difficult task to properly design a charging circuit for Li-ion batteries or one which will *PROPERLY* charge NiMH batteries without destroying their useful life. Li-ion batteries are particularly important to get right, as they present a dangerous situation, which is why no one is telling you on the Internet how to do so.
The project would be far less dangerous if another type of rechargeable battery were used, such as the small lead-acid AGM batteries sold for this purpose. It would not only easily power your intended project now, but would do so with a more advanced LED string design ( sets of series LED s paralleled from the power source ) so a brighter and more useful light can happen.
This battery can easily be recharged from a regulated 13.5 - 15 V DC source made of your solar cell / panel and a fairly simple boost converter circuit. It can safely be left connected and unattended to charge without much worry, takes up little space and would power about thirty of those LED s for about an hour if I don't miss my guess. It also won't explode, catch fire or do other obnoxious things to its surroundings should something go awry with the circuit in your absence.
I know we all like to keep a junk box and use it for things to experiment with, but is saving a few bucks here and there worth damage to your home, loss of your home, potential injury to yourself and your family or even possible death in a house fire? Electricity is as fire, a fickle servant and a hard master.
Here is a link to one such lead-acid type battery:
http://www.atbatt.com/power-sonic-12v-14ah-sealed-lead-acid-battery-w-f1-terminal.asp
I hope these suggestions will be taken to help you design something safer, better and more useful. Stay safe and good luck!