Harddisk LED Power Supply
There's probably a current limiting resistor on the "supply" side of the LED you are tapping from.

While you are unlikely to find an actual schematic for the board, perhaps you can trace it enough to figure out if there is a resistor on that lead, and tap power from the upstream side of it to supply to your led's through your own resistors.  

Or tap power directly from the output of the board's voltage regulator.

Or from the 5v line to the disk drives.

You will need your own current limiting resistors - you can work out the value from the LED specs, or you can just start too large and make them smaller until the LEDs are "bright enough".