I like the idea of the cleaners! The other day a vintage mangle (big ironing press) came up for free on Craig's List... if I only had the space.
Sometimes all you can do is start over with a fresh wash, while I don't usually use fabric softener, I might in this case. Do a full prewash and then be super careful about taking it out while still barely dry. From habit, I iron from the back side. For backing I will do a semi-quick press starting at one end and dealing with any snaggles, along the selvedge edge to the other end just the width of the iron, trim the other end and start down the other direction. Hopefully all you really need to do is the edges, but sometimes you do need to go into the middle or start in the middle to get the wrinkles out.
I also will move my ironing board so that I can use my couch or bed to support the fabric.
I've learned to get through the ironing by turning the music up loud and dancing while I press... still hate ironing but really, I get so much better results when I just go ahead and do it.