On the battery, make sure it is good as your first step. A good battery shop should be able to load test it either free or for a reasonable fee, you can test it yourself too.
The voltage check is easy but the hydrometer check provides more and better information:
Get a good, temperature compensated one, not a cheap plastic ball or float one. Here are two, the first I like and have used a similar one for years but it is glass and fragile, the second is recommended by several folks as accurate enough and sturdier. Either one make sure to flush them and get all the acid off before storing or they won't last as long or hold their accuracy as well.
Good one on Amazon: http://smile.amazon....DZI/ref=sr_1_17
Good reviews: http://smile.amazon....HMRU/ref=sr_1_2
On switching to LEDs I agree with Kirk, while it is cool to replace everything with LEDs it isn't cost effective. Look at the daily power use of each light (amps it draws times the number of hours you have it on) and only replace the ones that come in with big numbers. For us we replaced several lights with more efficient ones but only ones that saw more than four hours of average daily use.
LED shopping is complicated, it isn't just go buy a 12 volt bulb and plug it in!
First your electrical system isn't really 12 volts, it varies both up and down so any lights you pick should be rated to work with your system voltages. For someone on a dumb converter 11.0 to 13.8 volt ratings should be good, smart converter 11.0 to 14 something volts (read your converter manual) is needed. If you have solar then look at ones that won't have problems with your maximum system voltage. If you have a system that offers equalization (not just the voltage bump from some smart converters) I'd recommend disabling any automatic cycling and running manual cycles once you have turned off all the LEDs. Personally I disconnect all the RVs electronics when equalizing to protect them from the stress of the higher voltage.
Second is color, look for the CRI, color rendering index and buy as high a number as possible, low CRIs can look good but will shift some colors. If you use one over a makeup mirror you really want a high CRI, for non-critical lighting a lower number can do if the possible color shifting isn't an issue.
Third is color temp, orange to blue and different folks like different ones and sometimes different ones in different spaces. I like blue for reading, more red for computer and TV watching and mid-range for general task lighting. Blue (daylight) can disrupt your sleep cycle if you get a lot of it right before bedtime, red (soft white or cooler) can help your sleep cycle if you use it at bedtime.
Good CRI and color temp article with great pictures: http://lowel.tiffen....emystified.html
The prism photos are good here: https://en.wikipedia...rendering_index
The animated color temp GIF here is helpful too: https://en.wikipedia...lor_temperature
Now you can start looking at power use and lumens and picking the balance you want.