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It all depends...
Obviously, you are going to buy a 12V battery, as that is what the pump runs on. The real question is what capacity in AMPS or Amp-Hours.
If your pump consumes 1 amp, then a 7 amp battery (such as the sealed lead-acid types used for home.businesss alarm systems) will ideally yield 7 hours of operation. In reality, that batttery will probably only provide enough for 5-6 hours. But, if you switch your tank off periodically, or use a pwm controller* to limit current draw and water flow, you battery will last a lot longer.
Used Sealed Lead Acids can be acquired in San Diego for about $10 from INDUSTRIAL LIQUIDATORS on Convoy Street (across form the Dixieline). Typically, used SLA batteries are good quality because state law requires the batteries on fire alarm systems be changed every 6 months.
New batteries are 3 times as much at Batteries Plus, but you may desire new.
Walmart sells small 12V batteries for ride-on lawnmowers, motorcycles, etc for $20. They are traditional automotive-style wet cells that you fill with acid when you first open them (battery acid included).
For a charger, get one with an automatic charge circuit that detects when the battery is fully charged. Otherwise, you will eventually overcharge your battery and ruin it. (been there-done that).
*PWM- Pulse Width Modulation.... a circuit that limits current use in a DC circuit by providing a variably-controlled dc pulse output to power the dc device. The output is still the required 12V, but is current-limited to save energy. Kits are about $25 at Gateway Electronics (Chesapeake Dr. in Kearny Mesa). They are manufactured by Velleman.
Personally, I would minimize the electronics on the kayak and just use a manual switch to control running the pump. Just more to fail.
Hope that helps
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