I have two Latitude D510s. One will not power up. The batteries and power cords have been swapped and the problem stays with the one laptop. Pushing the power button has absolutely no response of any kind. Is there a fuse or something that can be checked.
Try discharging the laptop by removing the battery and AC Adapter then press and hold the power button for 15 seconds. Sometimes it can bring the laptop back to life. Otherwise I agree with ejn63, bad system board.
You do not need a new motherboard yet to get your Dell latitude D510 to power. If this system is similar to the Dell Latitude D600, then the system has a power board that plugs into the motherboard. Try taking the power board from the working Latitude D510 and plug it into the dead Latitude D510 to see if it powers up. If this does not solve your problem, have the DC Jack on the motherboard replaced by a capable technician. This demands that you know how to disassemble the laptop completely to desolder the current DC Jack and solder on a brand new one that can be obtained from Ebay. However, this is not a task for the average technicia, because it is tedious and demands a lot of care.
The new motherboard will set you back about $400-$600. You do not need to replace your whole motherboard because of a defective DC Jack.
I have full electronic lab capability, but wouldn't replacing the battery with a good one eliminate a bad power power jack. Assuming you are talking about the external power jack.