Your post doesn't indicate if you have considered resale tickets already or not; it's not "impossible" to get tickets, just incredibly expensive. Ticketmaster is more reliable than Stubhub, but Stubhub is a bit cheaper. If you do Stubhub, you can get two of the seats in the very back for about $2400 total, or "if you really love her", two seats in the second row (A) for $4600. For Ticketmaster, same/similar for $3400 and $4800 respectively.
The thing to note is that this is one of the dates that Lin Miranda will NOT be performing; I don't know if that would make a difference to you or your wife.
And of course, like others have said, the much cheaper option is to get tickets to another show and surprise her with Hamilton next year.
Hamilton Emotional Support Group:
https://www.facebook.com/groups/hamiltonsupport/
photopab said: "I asked my wife what she wanted for our 10 year wedding anniversary and she said "all I want is to spend the weekend in NYC seeing shows with you"... :)
We'll have a great anniversary no matter what, but if I could get my hands on Hamilton tix somehow, I'd be the #1 husband. :)"
Since you guys are celebrating your 10 year anniversary, why not see a show that created a big splash on Broadway 10 years ago --- like The Color Purple or Jersey Boys?