I can see your problem, I've had this VERY simmilar issue come to me before, ultimately it's your choice to snoop or not, This is where you make your choice, do you break a boundary by sneaking into your brothers personal life and reading his private facebook account data and relay it to your Inlaw? Or do you give the password to the girl, further erroding the privacy/trust bond between your brother and you even more than the previous mentioned idea?
Perhaps the right choice is to Stand Back and not do anything, That way you're not at blame for "things happening" however you could be blamed for "doing nothing" HOWEVER, you doing something requires you to access an account that does not belong to you, an account's private data belongs to the owner of the account (or facebook, I dont use it, fk them)

So, Do you Steal and Relay information "for the greater good"?
Do you Give away a Stolen password (i don't care how you got it) "for her"?
Do you stand back as an innocent bystander because you would have to break a technical law to "do something"?

You can obviously see wich of the three choices i picked.
Have a good gamble!

Just man up and ask your brother "Are you still with (sister in law here), are you seeing someone else?"