What's the cheapest way to resolve a slab leak?

The FIRST question is WHY is it WET under your slab?

Some dampness is normal but leakage is not..

Surface grading is a frequent solution.. (shoot for 1" per foot for 10') Make sure gutters are clean and the downspouts are routed away from the house.

If it is a seasonal water table issue, then you need a sump pump at a minimum, perhaps a perimeter (interior) drain system (also called a French drain)

AFA repairing the floor, the hydraulic cement answer is a good solution, but it doesn't cure the problem.. just a band-aid. If the leak is just seepage and there is no standing water, then its OK.

HD carries a SIKA product that is a self-leveling crack sealer that could also be used, use a backer rod to control fill depth

First are you sure you have a slab leak, because in my experience I've never seen a leak detect company zero in an area 12"x12", maybe 3 ft x 3ft and still be off by 3 ft. You can check this yourself to make sure by checking your meter. If you don't know how, all you need to do is turn off all faucets in the home, shut off the service, go out to the meter and look at the red needle on the meter dial, with all water off you have now isolated the water between the meter and your service, if there is no leak the needle will not move. If the needle moves there is a leak between the meter and the service. If everything is ok here then open the service so that pressure is applied to all fixtures in the home with NO water on. Check the needle again if it does not move no leak, if it does there is a leak. Also some water meters today are digitial, there will be a dial be you will be able to see the numbers move instead of a needle same results. If you are totally sure you have a slab leak, then the location of the leaked marked must require the removal some bath fixtures and counter tops from you discription. I don't understand what you mean by you have felt the heat from the floor tiles. Are you implying that this indicates a water leak? To you if this means there is a leak this is a new thing to me, never heard of it. Next how old is the home is it still under warranty? If so then the builder & the plumbing company who did the rough & finish work should be contacted. If you have lived in the home for sometime, this whole situation seems odd. The reason I say this is the home went thru about 4 to 6 inspections from the plumbing crew, plumbing supervisor in the field, builder and the city before a permit to occupie was issued. The chance of the water lines freezing under the slab is VERY remote, and a leak to just starting out of no where is also remote. The only possibility could be is a fitting under the slab, but wait, in 1st rough construction there is not suppose to be ANY fittings under the slab, on the water lines. The PVC pipes do have fittings but that is just for the soil drain, and if a fitting leaked you would never know it, it's not under pressure, the water lines are. So, my point here is that IF there is a leak, there needs to be a reason, if the plumbing company installed a fitting on the water lines they are possibly liable, why the crew leader who allowed a fitting to be installed was wrong and the field supervisor should have seen that and had the whole line replaced. It is worth the time to have all of this looked into with the builder and a person from the plumbing company, if they still exist. I hope I have not confused you too much, Good luck

Try chiseling out the crack enough to be able to force some hydraulic cement into it. It expands as it cures so will get very tight. If that doesn't do it, I would suggest that you look for the cause of the water under the slab; it's usually poor grading around the home, and/or rain gutter/downspout problems. Try the easy things first. Be cautious of the "waterproofing" companies; big cost, some are high pressure with outrageous claims to scare you into buying their fix, which is many times overkill.

if the leak is not extreme use dry lock on the crack and let dry 48 hours. this should stop the water seepage.