If you want, you can consider the failed socket dead and get a single stick of RAM for the system. If you want to use 2 sticks again, you will need to replace the RAM socket.
Before you even attempt a RAM socket replacement, check the northrbridge for a failure, and try using known good RAM. If you continue having problems with known good RAM and the memory controller is good, you have a bad slot. If it's a bad northbridge, then you will need to decide if the cost of getting a new chip installed is worth it or it's better to scrap the machine and sell it for parts.
Since you know the memory controller is good and using known good RAM still did not work, it's safe to assume the memory slot is bad. If the top slot is bad, the part will be hard to source (it looks like it's Apple specific or a uncommon slot type). You can find it, but it's not easy. If the bottom slot is bad, the part should not be hard to find since it's just a standard slot that the vast majority of systems use.
IMPORTANT: The top RAM slot with the serial number is easily damaged if too much heat is used. If you do not shield it off well or use too much heat, you are very likely to damage it during the repair. You should only use a hot air station with good temperature control for this job. THIS IS AN ADVANCED REPAIR.
The key to avoid damaging the top slot is to use a good amount of flux to avoid having to as much heat, as well as a good hot air station. You absolutely do not want to use a cheap hot air station, as this increases the chance of top slot damage.
Spend many hours studying what techniques damage the top RAM slot so you know what to not do. Once you do that, study what techniques work the best. If you get nervous at any point, you will probably get the repair wrong. Send the board to a shop who can do the repair for you at this point.
If you are comfortable doing the repair or are willing to risk damaging the top RAM slot, here is a link to the guide on how to remove the motherboard, so you can do the repair. MacBook Pro 13" Unibody Mid 2010 Logic Board Replacement