Preserve a LED ON after the activation of a reed switch
You might be able to use a thyristor for this. Once triggered they stay on until the current is interrupted by another switch. e.g.,







simulate this circuit &ndash; Schematic created using CircuitLab

You'll need to find one that will stay on with 10 - 20 mA going through it. Alternatively, you can make a thyristor with a PNP and NPN transistor. A web search should get you plenty of examples.