Update: for my first test the flashlight laster 24 hours and 7 minutes before it needed a charge.
Building a long lasting, rechargeable and powerful flashlight is easy and cheap. I only used the following.
An effervescent table bottle
6 LEDs taken from broken christmas lights
1 18650 used laptop lithium ion battery
22 gauge wires
Step 1: Building It
Cut a small hole for the switch. Run wires through the hopes to connect the battery, switch and LEDs together. I used an electrical crimp connector to hold all negatives of the 6 LEDs together. I then soldered 2 sets of 3 positive LED wires together. I then installed the LEDs in the bottle lid. I soldered everything together.
Step 2: Trying It Out
The flash like works really well. I wrapped the battery in cloth so it would fit snuggly and would not move in the bottle.
The light is very bright. I installed no focusing mirror so the light is very spread out.
To check the voltage or charge the battery pull the switch out a little from the bottle to expose the terminals. Connect before the switch and the positive of the LED. Connect this to a wall charger. This allows you to charge the flashlight and check the voltage of the battery with the LEDs disconnected.
Step 3: Charging
I added two small charging wires (one before the switch the other on the positive before the LED).
I can simply hook up alligator clips to the wires to charge. I do need to have a timer and check the voltage often so it isn't over 4.2V. I can easily check the voltage of the flashlight without opening it using the charging wires to make sure the voltage doesn't get too low.