I dug up and brought a Stalk Thornless Blackberry home in a big chunk of its own ground, and put in in my ground with a bit of manure compost on top about a month ago in Idaho, (Early May) and one of the big stems had leaves and broke off so I stuck it in a two litre soda bottle full of water and placed it on the back porch. Both get sun all day except the one in the bottle gets shade about 3 pm. Now,(early June), the one in the bottle looks healthy and strong, has a flower but no roots, and the one in the ground is just sitting there being a plant... no new growth. The mother plant where I got this from is blooming, so maybe its late because it endured a move? Will my flowered cutting root? Also, I'll take any pruning tips. (as I am asked to add tags to this post I see there is "rooting hormone" which I have not used. Is it necessary? Ill go read up on that. ;) )
Blackberries fruit on two year old canes - so the one that broke off is one. Your accident is similar to the one I had a few days ago - but as most gardeners know, blackberries are very eager to propagate, either from roots or tips that bend down and grow roots. My thornless one is manageable, but wild blackberries can drive a gardener mad...
So putting your cutting into soil should work, follow the usual recommendations of "three to four buds, keep them right side up, reduce foliage and tent to reduce evaporation". Remove the flower. (Depending on the length of this cane, you could even get multiple pieces.) I haven't heard of them rooting in water, but considering their "invasiveness", who knows?
For your plant, it depends (pictures would help...) what you have: Last year's canes won't really grow longer in their second year, but develop little bushy twigs that flower and fruit. In the first year, new canes will grow a considerable length, but look kind of bare, because they have only single leaves at those places where next year's fruiting twigs will be. The new canes are often a few shades lighter or greener than old canes.
If your transplanted just "sits" (but without wilting), let it be, it could be transplant shock. If you have only new canes left after your accident, they will basically "sit", once they are done with growing to their length - or they might get over transplant shock now, then push for more length. Second year canes are probably just a bit later. Again: pictures would get you a better answer.
As for pruning:
After fruiting / in fall, the second year canes should be pruned out to make room for new growth. Those pruned canes are propagating material if you want... In spring, you could cut the tips of those canes that will be fruiting, but I'm usually skipping this step.
If you have a runner, you can use the runner to make more if you "staple it" to the dirt at the place the leaves come out. I don't know if it will or not after you removed it from the plant.