I am trying to watch the TV but it won't work.
I am trying to watch the TV but it is not working.
The two sentences mean the same thing. I added the to the TV because we are referring to the TV, not the act of watching it. Also and is OK, however but is a better conjunction since the two phrases oppose each other.
The jar won't open; it is stuck.
The jar is not opening; it is stuck.
The first phrase indicates that you tried to open it, and then made this statement.
The second phrase indicates you are in the process of trying to open it.
The alarm didn't go off, even though I am setting it up correctly.
The alarm is not going off, even though I am setting it up correctly.
In the first phrase, since the alarm going off is a point in time in the past, you should use didn't instead, since we don't know what will happen next time.
However, won't would be OK if you mean that the clock was broken or not working correctly in general.
In the second phrase, it would have to be ringing when you made the statement. Like you were trying to stop the ringing but it wouldn't stop.