There may be some confusion here (i know i am

),
When you say stem what do you mean ?, the part in the pic is the steerer tube not the stem, the stem is the bit that the handle bars go through, from the pics above it looks like all you need are a new star washer (the bit thats broke) and maybe some spacers to fit between the stem and top of the headset.
The stem must be slack to adjust the preload on the headset, when you tighten the bolt in the centre of the top cap the stem is pushed down the steerer to load up the headset bearings, if the stem is tight on the steerer and you try to tighten the top cap the star washer will burst, i suspect this is what has happened here (takes quite a force to break a star washer), the top cap should only need tightened enough to preload the bearings, should not need a lot of force to achieve this, it's very easy to damage the headset by overtightening.
Those forks are correct for your frame, no need to swap the steerer or change anything (except the broken star washer)
Hope this is clear(ish)