I'm sorry you're finding yourself needing to start navigating this. I hope your experience of the world of infertility treatments will be short, simple, and effective.
I'm a mom after IVF (cycle 4) that was needed due to my husband's vasectomy (and failed reversal). Like you (though our situation was better defined and to some extent, anticipated -- we knew we needed to jump straight to IVF and that "just relaxing" or "giving it more time" wasn't going to help, but we also had NO IDEA starting out what a difficult time we *would* have), I started this process staring those dollar figures in the face and thinking there was no way.
I ended up spending far more than I'd have anticipated, and going into debt to do it. The only thing I regret is not spending more earlier -- time is not kind to female fertility (I was 34 at our first IVF cycle, 37 when our son was conceived). Of course, that's easy to say given that it did eventually work, but the guide that worked for me was asking myself each time, "If I try this and it doesn't work, will I regret having tried it?" Each time the answer was no, so each time I proceeded (I quit when that was no longer true -- not IVF attempt 4, but IVF attempt 6, as I'd hoped to have 2 kids. Even now, if anything, I wish I'd tried longer, not stopped sooner. But mostly, that I'd been willing to spend more earlier. Though, again, we knew we had to use IVF, so the "hoping to get lucky" strategy was off the table for us. Not really a blessing, but it did simplify decision-making.).
If you decide to explore the money-back options that others have mentioned (which apply to IVF, not IUI), pay close attention to what they require. They may, for example, require you not to take more than a certain amount of time "off" between multiple cycles (again, time is fertility, for women, and not in a good way), if more than one is needed, and/or have rules about when you must use frozen embryos (if you have any available from earlier cycles) versus proceeding with a fresh cycle. All these decisions are deeply personal, so you may not want the rules of a financing program to be the one deciding them for you. And while clearly the goal is "birth of a healthy full-term baby," it turns out that "making sure I get pregnant" and "making sure I have a healthy baby" are not always the same thing -- for example, US fertility clinics have very high multiple-pregnancy (twins and higher) rates, not to downplay the joy of twins, but even twins are much higher risk, including of really bad things like miscarriage, than singleton pregnancies. You want to be in a position to be the ones making decisions based on your own circumstances, goals, and history, not to be pressured into transferring "one more" to increase your chance of achieving a pregnancy -- though in all honesty, this can be difficult to do (make sensible decisions) regardless.
I have no real words of wisdom or strategies beyond those. The "will I regret having tried this if it doesn't work?" question was a very useful one, for me. I wish you and your wife luck. There are very good online support forums; if you'd like to be pointed toward some, please PM me.