I recommend not to use a mechanical relay. Those tend to bounce which will give multiple input signals to the keyboard.
Use something like a Photomos relay:
http://pewa.panasonic.com/assets/pcsd/catalog/aqy-dip-form-a-catalog.pdf (Photomos is a panasonic trademark - I'm in no ways associated with panasonic. There are similar components from other manufacturers, too). The advantage over a photocoupler is that a photmos relay is bidirectional.
The picture looks a bit odd: are there 3 ENTER keys, or is the ENTER key just so big that it covers 3 contacts? I assume the latter, since all 3 marked contacts are in parallel. Connect the output of the photomos relay across one contact (left-right in the picture you showed us). Add a series resistor of R=(10 V - 1.5 V)/10 mA ~ 1kOhm to the LED side of the photomos relay.
Observe the correct polarity for the LED! If you want to make the circuit fool-proof, add a bridge rectifier in front of the LED-Resistor circuit. Make the AC-side of the rectifier the 10 V input from your source, connect the DC side of the rectifier to the resistor and LED. Make the resistor 750 Ohm for this setup because some voltage drops across the brifge rectfier.
Regards,
Harald