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Convert patio lantern from halogen to LED

I have 4 rechargeable battery patio lanterns that I want to convert from halogen to LED. I'm pretty skilled with a soldering iron but don't know a lot about currents and circuits.
In the first pic, you can see the lantern with its JC G4 6v 10w halogen bulb, dimmer knob, charging port and pilot light. The charger is 7.2v 1650mA. The lead/acid battery tucked in the base is 4"x2.75"x1.75" and goes dead almost every year - hence, my desire to convert.

I'd like to find a warm, dimmable LED bulb that emits uniformly 360 degrees, and a suitable battery that will fit in the base. I assume that I'd need to replace the circuit board but have no idea with what. The easy way would be to put in a pack of three AA batteries, but I'd rather make it rechargeable if not a lot of hassle.

Being a newb here, I don't know if it's appropriate to ask for assistance on this kind of thing, so feel free to tell me to take a hike or to throw me a lifeline with your suggestions!

PatioLamp1.jpg

PatioLamp2.jpg
 
An 18650 cell might be a better choice of power.
They come as 2500mAh standard, avoid any higher as they are more than likely "fake".
They can be charged via a small module (last photo) available say Ebay from a 5v plug pack, some with solder input, others with usb type plug.
All could comfortably be housed in the base with some "modifying".
Only thing left would be to see if the 12v LED lamp shown below would fit your lamp space, otherwise choose another.

LedDriver.jpg



Charge Module.jpg
 
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