You won't necessarily see the other networks (some people think that hiding their SSID has security value, which is wrong but it does result in their network not being obvious to people trying to avoid interference).
Obviously there are any number of other sources of RF interference at 2400MHz that you won't be able to "see". It could be any of them.
Is it impossible to insert wiring into the walls?
Hmmm... this also doesn't fit the description of "15 feet" from you original message???
If you have an AP and a client that are 15 feet apart and cannot connect, I'm seriously suspicious that there is something wrong with one or the other of the AP/client combination!
What kind of signal strength indications do you get (when they are connected)? At that distance is should be maximum, by whatever measurement system you have. (Probably -30 dBm to -40 dBm.)
Things that can cause problems are tower cases totally blocking the little antennas. And since the antenna is so close to the case, the "block" can be in just about any direction (a reflection off the case can cause phase cancellation of the signal in the opposite direction, for example). There is no one simple cure for that, but you might try different antenna orientations on both ends. If all else fails, buy a short "pigtail" of maybe 18" or so, and move the antenna away from the case.
Also, if your equipment allows it, restricting the network to less than maximum speeds might make it more reliable too.
As for covering two floors with one AP, that gets right into the realm of "magic". Horizontal polarization of your antennas might be much better than vertical, just for starters. First that puts the main lobe of the pattern pointing up/down instead of having a huge null there. But another advantage is that if you are getting interference from another wifi network they are probably using vertical polarization, and by orienting your antennas horizontally you'll decrease the signal from the other network.
Going past that, if you are in any way a techie (you don't have to be a radio tech, but you do have to get a kick out of doing this kind of thing, or this is a no go idea), go buy a fairly directional antenna, and experiment. A $50 cantenna would do it, but any antenna with good directional characteristics will do the trick. And that trick is to learn just *where* the interference is coming from. Since you can't see it, that won't be so easy (unless you get a spectrum analyzer to hook up to the antenna, and then it would be a snap). But if you point your antenna to the east and get no interference, then point is west and do get interference... you get the idea! A little triangulation and you can figure out where it comes from.
Once you know where it is, use directional antennas and relocate your AP to a position where normal use means the interference is in a direction perpendicular to a line between the AP and the client. Obviously in some circumstances that will work well, and in others it may not work at all.
Another idea that is *really* out in "wierd's ville"... If for example you determine that all of the interference is coming through one wall and originates in the adjacent apartment, it might be possible to block it! I'm not an interior decorator, and can't tell you how to make something that would look good, but I'm sure that a creative artist could. But if that wall were screened off with any kind of metal screening (chicken wire, aluminum foil, whatever), it would do the job. One example would be to tack chicken wire to the entire wall and then plaster it. I'm sure there are creative people who could design something suitable, but I'm a radio tech and not an artist!