Add SATA Port to Motherboard?
You can download the free hwinfo32 app and run it.
Look under motherboard, and the SATA ports that are live and supported will be listed.
If there is a label next to the solder ports (like "SATA 1, SATA 2, etc.), then you can see if that port is active. If it is, you are good to go (as long as you are as good at soldering as you think you are).
I disagree with the naysayers above. While SATA connectors themselves are extremely cheap, many laptop manufacturers contract out the assembly of their motherboards, and they are charged by the component or by the solder joint. In those terms the cost of the connector is less trivial, and it makes a bit more sense to not include it. Motherboard layout is expensive enough that computer companies will use the same layout for multiple versions of a board, just without adding all the components to all of them, so those motherboard traces almost certainly lead to the SATA chip.
There's no reason in principle this wouldn't work. However, you are talking about very tight soldering tolerances and something like 20 solder points. It's going to be a difficult job and one that I wouldn't attempt (which is the only reason my laptop in the same situation still has only one SATA port!). You would also have to contend with the problems of increased heat inside the case and decreased battery life, but those are minor problems. If you can find the port and somebody who can do the job, then I think it would work.
It's likely that there would be a SATA port already in that spot if your motherboard had a SATA controller that could support it; connectors are cheap, chips are much less so. So, if you were to solder on a connector yourself, it probably wouldn't really do anything.