What are my options for supplying power to an M140 M-Type JAPAN 2+ W laser diode running off of a 5A constant current laser driver?
The laser diode looks kind of like an LED, but you probably want to be driving the thing with a constant current source set to somewhere around 1-1.3A or so (1.6A would be really pushing it), which your laser diode driver can do (There is probably a trimmer to set the diode current). 

The current mode driver will draw only however much current it needs from the power supply that might be either slightly more then the laser current setpoint (Linear driver), or less (switched mode driver or switching pre regulator), but in any case it will only take what it needs. 

At 1.3A the diode will probably be dropping about 5V or so (But this is temperature and phase of the moon dependent, which is why laser diodes of this sort are current driven), and so the INPUT power will be about 5-7W, of which somewhere between 1 and 2W will exit as coherent radiation, the rest will come out as heat, so you will need heatsinking to cool the diode assembly. 

If your driver is a linear current regulator then the laser driver will be dissipating your (11V supply  - the diode voltage (say 4.5V)) = 6.5V * the diode current, so will also potentially need a heatsink (There is a reason big lasers use switched mode drivers).  

Now a word about safety: 2W of laser is class 4 (in the old money) which means that both specular and diffuse reflections from surfaces are hazardous to vision (Never mind the direct beam). Bolt the thing down to an optical bench and use appropriate ANSI certified eye safety gear (and shielding with interlocks so that a stray beam cannot blind anyone ELSE) and you are all good, but please, nearly nobody needs a handheld class 4 laser. Gear at this power is NOT a toy, no matter what some seem to think.  

Finally, get yourself over to www.photonlexicon.com where the real laser nerds hang out, FAR better for serious advice then here.