Replacing the battery

It is worth repairing, you can either do it yourself if you can solder or take it into a guitar shop.

it you're replacing a passive pickups, you'll need to find a place for the battery (that'll be the hard part)
EMG specifies 25k pots, you might need those
plus a 'stereo' 1/4" output jack to switch the battery off when you unplug the guitar.
if you're not using the EMG routing bus (connector block thingie) yo want to wire it something like this

Could be a wholes series of things. Does it power on? Does your amp power on? Could be the mains transformer, could be faulty leads, are your guitar pickups active? In which case the battery for the pickups require replacing. Have you checked the volume on the guitar, fx unit and amp?

You can get these on ebay for not too much money-
Takamine TP 4T Series Acoustic Guitar preamp NEW
Takamine TP 4TD Series Acoustic Guitar preamp NEW Latest Revision

If you are not an experienced repairman and your guitar has any value, take it to a competent repairman to get it done correctly.
That said, here are some links to how it's done-

http://www.guitarrepairbench.com/electric-guitar-repairs/remove_guitar_frets.html
Learn About Guitar and Instrument Fretting Fretwork
Refretting Guitars
Repairing Replacing Frets on Guitars Prestige Guitars Blog

Either the batteries that you put in as new were already on their way out or you have a current drain that is depleting the battery. Use an ammeter to determine when guitar is off if there is any drain on the battery. If so, that is the culprit.

IF YOUR GUITAR has a built in tuner, make sure the tuner is off as it disables output while tuning... The other thing is to check if you have a volume control on the guitar that might be set too low.. Make sure if you are using 1/4 inch plug guitar cable that it is NOT a TRS balanced cable, as that will turn off the power in the guitar.

TRY REPLACING VOLUME CONTROL BUTTON.

i am not familiar with that guitar. But if it's a strat style guitar you can get the springs here....

http://www.stewmac.com/shop/Bridges,_tailpieces/Electric_guitar_tremolo_parts/Tremolo_Tension_Spring.html?actn=100101&xst=3&xsr=4684

You can try just adding a spring or replacing all of them....

Take it to a guitar shop and have them check out the soldering and connections within the guitar.
Martin has a record of cutting corners on their less expensive models.