After years of repairing cellphones, I'll give you the same advice I give I give everyone else contemplating what you are - DON'T!
Unless you have some experience opening a cellphone and removing the tiny connectors in it, you're going to cause a problem. Then you'll be here asking how to fix that problem. When you do, something else won't work. Etc., etc. By the time you're all done, you've spent days and hundreds of dollars - more than a new phone would have cost.
It's cheaper to bring it to the shop and pay then to do it. They've already practiced on unfixable phones, broken everything and know what not to do. So the fact that they're charging you 5 times the price of a screen to replace it isn't outrageous, you're paying for knowledge and experience - the same thing you pay a surgeon for. He has 2 eyes and 2 hands, just like you. But when he cuts into your head he doesn't kill you. When an experienced tech replaces your wife's phone's screen, he won't break anything else. (And if he does, he eats the cost of fixing it.)
(And it may not even be the screen, it may just need new glass. We really can't tell from the picture.)