Yes TIP120 is overkill at 60V and 5A but it will definitely work!
WRT your question about base current. The technical answer is to look up the graph of the "beta" of the transistor (often called hfe
or DC current gain). Multiple the beta (at your collector current) times the base current to get the load current. However, this method is not generally recommended. Historically transistor beta curves varied pretty widely per lot so you could not really base a commercial circuit on it. But you can do this if you tested each transistor and eliminated out-of-spec ones (i.e. hand-picking). Today things might be a little different; I've done a bit of hand-picking recently and it turned out that all the transistors were right on the money -- none need elimination.
But the real answer is to use a constant-current circuit. This circuit uses feedback to control the current, so is immune to changes in beta and also LED voltage.
There are lots of choices, including DIY or my Lightuino LED driver board www.toastedcircuits.com
, but fewer for high voltage or high current. I recently posted a constant-current circuit for that here: http://arduino.cc/forum/index.php/topic,62893.0.html
Or at 40mA you'd probably be fine with a current limiting resistor.