Sounds to me like your two-element heater (the common design) has had the bottom element die - so the only hot water is at the top.
The reason you are getting very cold water when you drain the tank is the filler tube (connects from water inlet at top of tank to the tank bottom, commonly in a spiral at the bottom to promote water mixing) goes to the bottom of the tank for that water to be heated at the bottom element, with the heated water rising the top due to convection and then reheating hotter by the top element, ready to go out the top exit pipe for use. I would suspect the initial water that came out was warm for a second or two, before the cold refill water came in through the filler tube.
Normal vendor would be a Plumber to replace the element - not a tough DIY job either, as long as you are not afraid to remove and replace electrical connections at the element (with power off, of course) and are confident (on most models) in your ability to get a good thread seal where the element screws into the tank, because most elements stick right into the water, not into a well or recess. Some use a gasket rather than threads, so easier to get a watertight seal.
Cost for plumbear to diagnose and replace - element about $10-25 each - though commonly come in two-pack and are replaced as a pair on the assumption if one failed the other is probably not far behind, so $20-50 parts in that case - and likely about minimum service charge for the labor of typically $75-150 range except more in a very few very high cost areas. Possibly more if a large tank because of the time it will take to test for hot water in a large tank, though many plumbers will leave to another job and come back a few hours later at no added charge (or just tell you to call if not working right) to check it is coming up to temp correctly and temperature setting is where you want it.
