This strange Irish name is read very simply - [tay].
The surname is originally French, and should be read as [tange']. But in America it got americanized, and now the name of this NHL player is pronounced now as [tangway].
Another example of a "-ham" suffix, where "h" is mute. The word itself should be pronounced like [tottn em], with a little break before [-em].
The second name of an Argentinean defenseman is stressed on the first syllable.