How to transplant orchids embedded on a tree branch

[Katroushka]

I live in Costa Rica and received as a gift a couple of years ago a thick tree branch covered with orchids. I love gardening but I've always felt slightly inept when it comes to orchids. However I've kept it alive and thriving, new little bulbs appear all the time to replace those that die off, and even though it's only bloomed twice in 2 years, I'm quite attached to it. I have it hanging indoors half of the year as it's very dry, windy and hot where I live for 6 months, daily sprays, monthly fertilizer, and weekly soaks in the shower; and the other six months I hang it in a semi-sheltered spot outdoors to get tropical rain. To my dismay the orchid branch fell from a 2nd floor balcony yesterday, snapping the branch in 3, and revealing rot inside. So it's time to move all those little bulbs and leaves to a new branch (or other perch) and I haven't the slightest idea how. The roots run all over the branch, intertwining with one another, and I don't want to damage anything by lifting them off, given that the fall was stress enough on them. Help!