Your best bet for a good finish relies on doing three things:
- Making sure that the foundation for the extension is well compacted and onto good material
- Cut back and scable the joint with the existing 10x10 slab so that there is a good bond when the new concrete is cast.
- Drill and fix dowel bars into the existing slab across the joint with the extension.
The number and type of bars will vary depending on the load from the new shed but you should be fine with something like 8mm (3/8") bars at 200mm (8") centres.
You should probably note that this will give you a full structural connection and so may be too much for your particular problem unless you want to be absolutely sure that the slab and the extension don't separate.
You can install an interior weeping system similar to what you would normally do on the outside, but without the waterproofing.
Start by jackhammering out about two feet of floor around the edges of your garage, as close to the walls as you can get. Dig down until you reach dirt, and if that's not at least two feet down, keep going. Now, put down a couple inches of 3/4" crushed gravel, and then install weeping tile (it's not tile anymore; it's 4-6" corrugated plastic drain pipe with a nylon mesh sock over it). Make sure the sock is continuous across any joins you make in the weeping tile for corners etc. This is a good time to direct those drain holes down into this trench, where the water they drain will flow into the weeping tile. You'll also need a sump, or a connection to a downhill storm sewer; direct the weeping tile to this sump pit or the storm drain. While you're doing all this a center drain can't hurt too much; you can tie it into the weeper wherever it's convenient, just make sure the garage floor and the drain flange are level, or that the garage floor slopes slightly to the drain.
Now, you can install waterproof sheathing on the interior walls contacting the concrete. Normally this product is designed to go outside the foundation walls, but in cases where that's infeasible it can work this way too. The idea is to trap moisture that weeps through the wall behind the sheathing, where it will then be directed down the wall and under the slab to the weeper. So, you'll install the sheathing, which should have an air space between wall and most of the layer, and make sure the end is down in the trench by the weeper.
Backfill the rest of the trench with 3/4" crushed gravel up to the slab footing, then patch the slab with new concrete. Your garage should stay much drier.
Best Answer
Just a gut feeling, but based on what I've seen that will hold a brick patio flat for a long time, two or three three inch lifts of packed crush under a 5" slab with 10mm rebar at 12-15" should be adequate. If you want something set on dirt to stay flat and the way you set it a long time, multiple layers of tightly packed gravel give a flat, long lasting high strength base with good drainage for the brick or slab on top. I don't know enough to be sure of slab thickness but 125 PSI seems like nothing. It's about 0.86 MPa and Wikipedia says regular strength concrete is good for at least 10 MPa. If you do a solid base, you can probably even get away with a 4" slab as a hot tub has fairly wide weight distribution and what usually breaks 4" concrete is a wheeled vehicle that concentrates its weight.
Fair warning I'm only aware of electrical codes for hot tubs and my solution might be overkill, but I try to build things to last. Gravel is relatively cheap, digging, placing and packing are hard work, but you only have to do it once.