Mývatn Nature Baths are the only paid baths where we have been. The structure is really similar to the more known Blue Lagoon (near Reykjavik), but the price is much cheaper and the place is not so touristic, because it is in the North of Iceland and definitely not a place where you can go and come back in one day from the capital (link to google maps, https://goo.gl/maps/dQxR9qsFAMDA15oR6). To get in it costs £28-£32 depending on the period of the year, but if you are a student you can get a discounted ticket at the price of £19/20.
You can stay in the structure for as much as you wish, beside the mains bath they also have a sauna and a smaller pool with even warmer water.
One thing to remember is that because of the minerals contained in the water (that are supposed to be good if you have any kind of skin condition) it is better to take off your jewels, rings and glasses because they could get ruined. Also one thing I would like to have known before is that they do have showers but the shampoo and conditioner they have for everybody to use is not of the best quality from my point of view and this can be a problem for people with long hair because the water of the lagoon makes them quite dry and frizzy.