If the backlight issue happened immediately before you noticed the battery problem, the two things may be related. If the battery got really hot, it probably had a short-circuit, and the heat may have affected the backlight.
Replacing the battery is straightforward, AFTER removing the two white plastic end pieces. They are held in place by internal clips AND some adhesive. Not easy to remove the first time. I have this tool
http://eshop.macsales.com/item/NewerTech/TOOLISESAMO/
which is very useful to create an initial gap between the plastic and metal casing. It slides in easily between the metal and plastic. I was not able to open a gap using just the plastic (nylon) tools, and using something more blunt like a small screwdriver would have caused ugly marks. Once I opened a small gap, I wedged the plastic tool into the gap and slowly pried the plastic piece off the rest of the way.
OWC has a HOW TO video linked to their replacement battery
http://eshop.macsales.com/item/NewerTech/BIPOD650MM/
which advises heating the end pieces to loosen the adhesive; I did not do that. I bought a replacement battery on Amazon, and it is much higher capacity compared to the original; it seems to last "forever" on a charge. I also replaced the tiny hard drive in my iPod mini with an SD ("SDXC") card on an adapter. The stock "micro hard drive" is actually the same physical size (with same connector) as a standard Compact Flash card (most CF cards are thinner). You can use a CF card directly without an adapter. I used an SD card (with an SD-to-CF adapter) because it was cheaper. My "4GB" iPod mini became a 64GB iPod mini, large enough to hold my complete iTunes music library.