You may not find an actual Mallard freewheel tool but there's only so many patterns and one of those patterns will work. Or just take the wheel into a bike shop and ask them which it is. Or go to www.parktool.com and look at the pictures of their tools to select the one you need.

Your wheel has a solid axle and his has a quick release axle. The difference is that your axle uses the nuts to hold it on and his uses a quick release skewer to clamp it into place. The skewer is missing on his wheel. But again bike shops could help you with this aspect.

You say that on his wheel the gearing doesn't move freely. Can you spin it for multiple turns and it's just gritty or does it only turn a couple of times and then locks? It's hard to tell from the picture because the hub is so fuzzy but it's very odd that there's only two sprockets on that hub. It may well be that it's some funky two speed fixie thing as mentioned. If so then the gears would turn a few times on the threads and then lock if the lockring is loose. If they keep turning then it's got a freewheel of some sort that you can swap out. But that means you need TWO tools unless you're lucky and the one freewheel tool fits both freewheels. You'd also need a skewer to get this wheel back into use since as mentioned it's not likely that the cones and spacers will fit between the two wheels.

There IS another thought. Remove your freewheel and replace the spokes that are broken along with any others that are badly loose and then have the whole wheel properly tensioned and trued. From your post it's likely that you should get this done by the local shop or find someone that has done this before so they have the right feel for it or bad things can easily happen. But since it sounds like there's nothing wrong with your wheel other than a poor tension and true doing this would make it fine again and likely better than it was before. And get the front treated the same way since very likely it's poorly tensioned as well.