I would have the battery load tested before you do anything else. Even a new battery can be bad. Any auto parts store will do it for free, I bought my own since we got so much crap with batteries. After that, remove the selenoid and starter and have them tested. Remember, a battery can show good voltage, but under load will do nothing. Start with your batery and work your way towards the starter.
If I'm understanding correctly, even when you jumped the battery you only get a click when you engage the starter. If that is true then I would suspect the starter itself is bad but there are a few things to check first before we blame the starter.
As you might imagine, it takes a lot of power to turn over an engine and during starting all of that power must be supplied by the battery. Without getting too technical... in electricity power = wattage and wattage is determined by the voltage times the current. All we've got to work with is the 12 volts in that battery so to get enough watts to turn that motor over we need current and lots of it. In some cases several hundred amps.
In order to get a couple of hundred amps out of a 12 volt DC battery we need some very thick wires (the thicker the wire the lower Resistance). That brings us to the number one failure point in any starting system... the connections from the battery to the starter motor. These connections must be meticulously clean because even a very small amount of extra resistance will cause a huge drop in the current and that means loss of starting power (watts).
The first step in troubleshooting a no cranking problem is to check the connections at the battery and at the starter motor. Make sure they are tight and meticulously clean. (no, not the outside, the inside where the metal of the wire meets the metal of the stud or connector. Don't forget the thick black wire (negative from the battery). This black wire will connect the negative of the battery to some substantial metal structure probably on the frame. If these connections are tight and clean then we can move on to the next step... the starter circuit.
The tiny connections inside your key switch could never carry the huge current loads required to start an engine so we need some other switch mechanism to close the high current circuit between the battery and the starter motor. That job falls to the starter solenoid. The starter solenoid consists of a heavy set of contacts that are activated by a small relay that is energized when the key switch is rotated to the start position. If the key switch is energizing the starter solenoid then you will be able to hear a "click" when you turn the starter key. The fact that you hear the click means the circuit is functioning but we don't know about the high current contacts that the starter solenoid is supposed to engage yet. We only know that the circuit from the key switch to the starter solenoid is functioning normally.
Inasmuch as most late model starters come with the starter solenoid as an integral part of the starter... we can be relatively certain that if we are hearing a click when we turn the starter key... and we know the battery has enough power to turn the engine over (we jumped it) and we know that the thick wires that connect the battery to the starter are good... then there is only one thing left that can cause the engine to not turn over (assuming that there is nothing jambing the engine which is preventing it from turning)... the starter motor is bad and needs to be replaced.
However, there is one more check we can perform that will further prove that the starter is at fault rather then something else preventing the engine from turning. If the starter is good and the battery is good and the thick wires connecting the battery to the starter are good and we hear the starter solenoid click when we turn the key but the engine still won't turn over then hold the starter engaged for about 5 to 10 seconds and then let the key switch return to the "on" instead of "start" position. Now go feel the thick red wire on the battery. If it is hot to the touch then something is preventing the engine from turning but if it is cold to the touch then we can be sure that the starter is bad. (as long as all the other conditions mentioned above have been met).
Note: Under certain very special circumstances it is possible, but not very likely, that the thick red battery wire could get hot to the touch even when the starter is faulty (instead of something preventing the engine from turning) but this would require that the starter itself is jambed (or stuck) or it has a shorted starter winding and although not impossible it is an unlikely failure point. Starters can and do short but it is very likely that they would also overheat and perhaps even burn or smoke when energized and that would be hard to miss.
Sorry if I've been over simplistic or too long winded but I have no way of knowing what you may or may not know about the starter circuit. I hope this helps you.
