Water Heaters - not enough hot water

I have a typical problem of not enough hot water, especially in the morning, and am hoping for some ideas. The problem is I lose hot water after less than ten minutes, I am unable to fill the tub more than 2-3" without it being cold, and it has very poor recovery time. In the morning it's worse. If I run the laundry with hot water about an hour before I need to shower, then my shower is much better.

The water heater is about one year old 40 gallon Home Depot gas heater. Unfortunately I've let it go beyond installation warranty coverage. It replaces a 40 gallon Smith water heater that was able to keep up reasonably well, even with an old shower head.

So far I've tried:
- replacing the showerhead with a new 2.5 gpm one that has significantly less water flow than the old one. Helped a little but not much.
- flushing the heater. I didn't see any sediment and it didn't make any noticeable difference.
- turned the thermostat all the way up. This helped a little but not much. Now the sinks have scalding water, and I have two little ones to worry about.

What else can I try?

Check to make sure the hot/cold lines were connected correctly on the water heater. Look on top of the water heater where the lines go in/out. There should be a "C" and an "H" stamped into the sheetmetal near the nipples. Turn on the hot water somewhere in your house and then go feel the pipes at the tank being careful about high temps. The pipe connected to the "H" side of the tank should be hot and the cold side should be cold.

Those are correct, and other faucets work correctly.

Of course, we have only the one bathroom, so nothing else realistically uses that much hot water.

A 40 gallon tank will give you 28 gallons of water for a shower, at a temp of 20º below setpoint. That is about an 11 minute shower, depending on just how hot you have it set, and how hot a shower you take.