I fixed up my doorbell to be much less sketchy. Now, a 555 timer constantly runs at a visible duty cycle (around 17 hertz), driving a 2n7000. When the button is pressed, the voltage source is connected to a 1000uF capacitor and powers the LED. When the button is released, the capacitor discharges through the LED and another resistor to make it drain faster. I probably should have a resistor in series with the LED, but the LED I'm using is tough and a good brightness without any resistor. I'm relying on the fact that the voltage source has some internal resistance for current limiting. Here's the circuit and the new setup:
New setup outside, aluminum tape as wires. I'm great at neatly aligning tape aren't I?
I had to reinforce the connection on the inside with hot glue since aluminum tape isn't very structural.