We did our IVF at the Utah Center for Reproductive Medicine. It is tied to the University of Utah hospital. Our latest round of IVF was around $13k (including drugs). The sperm retrieval was an additional $1200. Sperm retrieval was done with a process called a tess-e (or testicular biopsy). An urologist make a 1-2 cm incision in the scrotum and take a little bit of testicular tissue. From that tissue they can extract enough sperm for several round of IVF (they cryo-preserve it for a small fee). I was drugged up for this process so it isn't bad (it is outpatient), but you do swell up a bit afterwards and have the general feeling of being kicked in the groin for 1-2 days. Not the intense immediate pain but the pain of being injured. Sitting on a bad of frozen peas, and pain meds make it tolerable, but don't plan on doing much more for than sitting and short walks for a couple days.
Your poor wife will have the worse part of the deal. The Dr uses drugs to stop her cycles, then use other drugs to create a hyper cycle so her ovaries will mature multiple eggs at the same time. This can be a bit emotionally messy (I've heard it compared to menopause and really bad PMS at the same time). The Specialist then retrieves those eggs with a trans-******l needle. Then a lab uses the ICSI process to put your sperm into her eggs. The eggs mature a few days and hopefully you get 1-2 mature enough to implant (Extra embryo/blastocysts can be cryo-preserve) Then her eggs are transferred back to her uterus, and she takes more drugs to trick her body into thinking she is pregnant.
If you are lucky you have a positive pregnancy test in 2 week followed up by viability ultra-sound in 6 week and all kinds of joys and stress that comes with pregnancy and parenthood. If you aren't successful you feel lonely, heart broken, and a little bit mad at God for a couple months. We've had both success and failure. The failures are really tough even when you already have a couple kids. The successes are phenomenally life changing in the best way. I wish you nothing but success.