have you tried bending the pins back? If they aren't broken, it might be worth a try first. Shorting with other pins might have blown things further back, if power has been turned on since this happened.
Do you have a wiring diagram you can post here? Is the ecm triggering the relay coil? These usually have issues with the pump control (usually brought on by human error or as simple as dry solder joints). If you can post photos of inside the ecm on both sides of the board, we should be able to see it.
I used to buy ultrasonic cleaners for our company. Does it come with built-in heating? That type are particularly good, over those that don't have it. The key thing to look for when buying an ultrasonic cleaner is not just brand or build quality or materials. You also need to check if the product is supported with spare parts availability - especially the sensors as they are near-impossible to get hold of at anything like a reasonable price. Now, the single most important thing to look for in an ultrasonic cleaner, is quite surprisingly, its size compared to the parts you are putting in there. You need to match the cavity size to the parts size EXACTLY. If you think bigger is better, you'd be wrong! The effectiveness decreases considerably to almost nothing when you have tiny objects inside a large cleaner. Most buyers do not take this into consideration & blame the product or the cleaning solution, etc.
It might pay to get a code reader on it & let us know what it tells you. Sounds like the transmission control computer is giving problems. Open it up & have a look for the obvious capacitor leakage problem with this one. These computers have a history of SMD chip resistors going open circuit, which is impossible to spot, but easy to measure with a multimeter. I've personally repaired hundreds of similar systems over the years.
From experience, I'd say it's a surface mount capacitor. You need to take it out of circuit & measure its value or else get a schematic & look up what ratings it is supposed to have. They are quite cheap to buy, so try getting a new one, as pilfered parts can be difficult to take the solder after removal. Alternatively, measure it using an in-circuit capacitance meter. It looks to be only used for filtering, so I don't imagine the exact value is critical. If I had to guess, I'd say it'd be around the 100nF mark. Such a part doesn't usually go faulty, so maybe it just needs a re-work?
Batteries generally perform better when warm & worse when cold, but I wasn't expecting it to die so suddenly when exposed to cold. Whilst it's still lasting a few hours, spray some freeze-it spray (available readily from shops like Jaycar), onto a section of the board to try to simulate the cold to see if you get the same effect. Try a different section of the board after it recovers each time to see where the problem lies exactly. If it was me, I'd replace the battery with a pack I can make from cheap, common batteries, just like they showed us in our country's only Electronics Magazine, Silicon Chip some years ago. I may be able to look up the article, if you wish. Update (09/03/2016): Ah! I should have read your initial post more closely... You've had the phone for 1.5 years & six months after you got it (new or second-hand?), it started giving problems. I guess you've heard about Apple's new laptops which explode when you (or anyone who's not an authorized Apple repair centre), try to change the...
Check the winding resistance of the suspect motor against the others - open circuit. If it's okay, then you need to trace it back to the driver & change it if it checks out faulty or just order another controller board.
Can you tell us how you've checked the fuse & relay? What equipment do you have available？ The power for the pump comes through the Fuel Pump Relay. Can you tell us what voltage you have on each pin on the relay, please? The pins should be numbered on the relay itself. Do you have battery voltage on one side of the relay switch & not the other side? If so & the coil is still good & you cannot hear / feel the relay clicking on crank & the wiring is good (check for breaks as well as shorts to other wires / chassis), then it's likely the engine computer isn't switching the relay in. Trace the pin through to an smd transistor or mosfet & change it or just get another compatible unit from the wreckers (usually quite cheap).