OK, being an electronics service repair man, here are my soldering tips:
1) NEVER use flux when soldering electronics !!!! Flux is for soldering water pipes !!!
2) The flux is inside of the solder. Because the solderings on that keyboard PCB are probably reasonably old, the flux on the outside of the soldering is gone. My pro tip that I learned at school: Take some solder and "re-solder" the old solderings. The flux in the new solder you hold to it will make the old solder go fluent. USE ONLY A TINY BIT OF SOLDER to do this !!!! Just enough to get it fluent, then remove the new solder immediatly, and remove soldering iron as soon as the old solder is completely fluent.
Work this way on all the solderings you want to make.
3) Pre-solder (don't know how to call it really in English) the wires you want to connect to the old solder spots. Your wires are stripped down WAY too long by the way, but actualy that is good now. First, twist the wires round with your thumb and indexfinger, so that you make it a "single" core. Next take some new solder and hold it to the wire, and hold the soldering iron to the wire (not the solder), You can "play" with this a bit if the solder doesn't start running, but the basic thing is that you let the solder do the work. Actualy it is a physical thing called "capular effect". Because of the air between the wires, the fluent solder gets drawn in "automaticaly". This is a basic thing you need to understand and apply, let the solder do the work, no smearing etc.
A good solder spot is never a ball, but always a "fuji" (yes like Atari's). So NOT like this:

You get these balls by using TOO much solder.
OK, when you prepaired all the wires this way, cut them with a good (sharp) pliers, and let the length be minimal, so minimal that they cannot short-circuit to eachother or the PCB's other solderings. I'd say 4 mm is ok.
Now WITHOUT using any new solder, hold the wires to the "old" solderings and use your soldering iron. Remember, don't solder too long, never more than 5 seconds.
Because you "renewed" the old solderings and because you pre-soldered the wires, it should now be easy to solder them together.
However, soldering wires in this way to a PCB it is very likely they get loose. Contrary to what a lot of people think, in electronics, a soldering is NOT a mechanical connection ! This is why a part is resting on a PCB, etc. SO you will have to fix the wires to the PCB in some way, so they won't move. A glue gun could be an option.
However, I'd still suggest to use a connector. My tips work with connectors too by the way, first solder the connector lip, then the wire, then without adding new solder, solder them together....
Jeeez, there's a lot of "solder" in this posting

More soldering info:
http://www.epemag.wimborne.co.uk/solderfaq.htm