If you are still having problems try to be a bit more specific about the what the circuit does or doesn’t do, including scope shots (assuming you have a scope) of the output of U1 as well as the dis/charge cycle of C1. And the exact part number of the 555 you are using.
As for the barrel jack, pins 2 &3 for a switch that is usually used to disconnect an internal battery when plugging in a DC adapter.