As above, and in particular take out the operating levers, two piece things, item 7 in the link below, which Jordie particularly mentions, and clean them, lightly lubricating the rivet which joins the two parts. Often that vital pivot point siezes, with the result that only one shoe is being applied to the drum. If you can't get them working freely, throw them away and get a new pair. Fit new clevis pins, as the originals are inevitably worn.
http://www.somerford...page=page&id=60
You should not need to fit Minifins, which are very rarely justified on a road car, as a cure for problems, as if the standard system is not working correctly, the Minifins will not help in the slightest. All they do is to marginally reduce unsprung mass and improve brake cooling, but only in the most extreme cases of competition cars is there a thermal problem with the rear brakes. Minifins, the delight of the various Mini suppliers, because they are profitable, are a solution to a non-problem, like quite a few other mods too. Having said that, a genuine Minifin is a well-engineered item, and does no harm to the car, even adds value and looks good, however many similar drums on the market these days are dangerously defective counterfeits which suffer from serious problems such as the cast iron liner loosening in the alloy. Keep genuine Minifins, but be aware that they are neither helping nor hindering the basic problem.
All you need to do is to give the standard system the maintenance that it should be having regularly, and the brake efficiency will achieve the MOT requirement. Not, admittedly, by a big margin, because braking efficiency at the rear of the Mini is severely limited due to the lack of weight in the rear of the car.
None of my Minis ever failed the MOT on handbrake, because just putting bits into their normal condition is sufficient to get a pass. With the mileage I used to do, the drums were off at least twice, maybe 3 times, a year, so everything was clean and well lubricated. It doesn't take long, once you have done it once and become familiar with the mechanism. (One at least was abysmally bad the first time I took the drums off, as it had been neglected by the previous owner.) I think that so many people have problems now, simply because they tend to do low mileage, and the drums are off maybe every 3 or 4 years, which is not enough, considering how the pivot points are prone to rusting. That goes for other mechanical problems too. Use it hard, service at the recommended mileage, and it will tend to be more reliable than if it is rarely used. So it is helpful to do more of what Minis are for, and drive the thing, moderately hard.