Single Package Transformer
You don't state what sort of space it is: If it's a PCB then something like 

https://uk.rs-online.com/web/p/embedded-switch-mode-power-supplies-smps/6675760/

would give you 500mA of 5V (and would be superior to a dirt cheap usb adaptor), you'd then just need a resistor from that to power your led.

If your led is well enough insulated you can power it directly from mains AC: you'll need a resistor to keep the forward current down, and a diode to keep the reverse voltage down, or you can buy a pre-packaged LED with all the components inside eg.

https://cpc.farnell.com/camdenboss/cbpi22-230-rd/22mm-indicator-led-pilot-light/dp/MC02599

You could also use a neon indicator like the following

https://www.digikey.com/product-detail/en/visual-communications-company-vcc/1050QA1/1050QA1-ND/3150873