Unlimited power! But how to put it to good use? (motor torque)

Well not exactly unlimited power, but a lot of power! My wifes lower back is worn out so she can't walk long distances and because not everything is reachable by car we have a foldable disability scooter to overcome that problem. Unfortunately the batteries needed replacement, quite expensive.

The good news is that I now have the old two 12v traction batteries. One can only hold about 60% of it's charge, not of much use. The other is 90%, definitely good enough to use and since the battery is about two times as small as a regular car battery it's perfect for a robot!

I want my robot a bit bigger. Not too big, I was thinking about LxWxH = 30x30x40 cm or something like that, so the extra weight of the battery isn't an issue. Weight is of course a bit hard to predict, since I haven't build anything yet, but I guess the total weight will be about 5 to 10 kgs.

Speed is not really an issue. I'm thinking about a max speed of 2km/h. Power is another story, too large motors use more power than needed and besides that: I don't want a robot that runs through a wall instead of against a wall in case object detection is a bit flawed.

I'm not sure about the size of the wheels yet, but they will be either 10cm or 15cm in diameter. If I take 15cm then the robot travels about 47cm with one revolution of the wheels, that means that for a maximum speed of 2 km/h the wheels will turn at about 70 rpm. The geared motor shaft will be connected to a toothed belt so that I can mount the wheel shafts in ball bearings.The gearing of the toothed belt depends on the motor, but about 1:5 sounds nice (for the encoders) so in that case the motor shaft has to turn at about 350 rpm. Not exactly rocket science, more of a guide line.

Now that I've got the maths I need the power and here I'm a bit clueless. The geared motor I'm looking for is reasonably silent, has a speed of about 350rpm at the output shaft and it should be affordable of course, but how much torque do I need approximately at the output shaft of the motor?

(Motor recommendations are welcome of course and an explanation how to calculate which motor to use will be more than welcome, I like to learn.)