Power supply for electric scewdriver

[LSWRArt]

The battery is almost dead on my 9.6 volt Black and Decker electric screwdriver and now the charger has packed up.

I could buy a new screwdriver, but I always have the problem that the batteries are never charged when you want to use it and after a couple of years use the life / power fall off, so I decided I would rather make a mains powered PSU and drive the screwdriver directly off this.

I have a very old bench power supply so set this to 10 volts and tried to power the screwdriver off this. The volts dropped from 10 to 4v and I was measuring a current draw of about 6 amps, so it obviously needs something pretty meaty to power the screwdriver.

Living in France there is no RadioShack and I have to order everything mail order, so I need to get the design right first time.

I thought a mains transformer with a 12 volt, 10 amp secondary, feeding four 10 amp diodes to make a bridge rectifier should work.
Is that likely to be enough power?

It is so long since I did electronic theory, but I seem to remember I need to divide 12 volts AC by root 2 to get the DC equivalent voltage, so 12 volts AC would give 8.5 volts DC?

Do I need a smoothing capacitor - if so how big? I think the problem then is that the capacitor would boost the output to the peak AC volts of 12 volts, so the voltage would be a bit high for the screwdriver motor. I cannot easily buy a 9 volt, 10 amp transformer.

Many of the mains transformers available here are 12 - 0 - 12 volts output. Can I use both outputs and two diodes off the two 12 volt legs, then join the outputs of these two diodes together to give the same DC output voltage?

It is more than 40 years since I did any electronic theory, so struggling to remember even the basics!!!

Thanks for your help
Art