I honestly think you need to call in a team of professionals to do the job. You can either pay them now, or pay for more building repairs later. I opt for the former.

If the stems were thick enough to need a chainsaw could you use SBK on the stumps?

Is this photo now or prior to your removing the ivy? I've never seen ivy grow that much in four months!

I think the topgrowth we're seeing has been there years - it just hasn't died back yet since you cut it all off at the base. When you say you used a 'good weedkiller', unless it was something like Paradise, which kills everything in the vicinity and renders it unplantable for up to a year, your ivy will grow back. Each root should be treated using SBK.

The immediate problem is the topgrowth. Unfortunately, its likely that growth has rooted into the walls and roof of those buildings, and if you try to remove it all by pulling off growth that's underneath, against the walls and roof, it will cause damage. Whilst superficial damage can be repaired, those buildings look rather rickety, and you might find parts collapse completely. If you didn't want the buildings, or were building new ones, then employing someone to clear the lot using a digger or an earthmoving machine, knock it all down, then remove the debris to a skip or something, is the easiest option. There is no easy option if you want to save the buildings, I'm afraid its a case of cutting it away till you get to the underneath and see how well its rooted into the structures.

If you can afford to wait though, once severed from the root it will die off and go dry, though this may take several months. When its quite dead and dry at least some of it should lift off without too much trouble, but it won't undo any damage already done and bits of shed might lift off with it..

Never thought of the fact it was going to takes months to die back... should have Bamboo when Bulba cut the ivy off at the root that was growing up our wall it too a while. Stera, I think there is simply too much ivy there to wait for it to die back having been cut off... needs, very carefully, removing in my view.

Ivy that has got this big and has been cut off at the base does not mean that it will all die back, why? Well because where the ivy has sprawled and strangled every nook and cranny on the roof etc over many years then it will have rooted up there too, this can be dealt with though, over the years I have had to deal with many ivy situations like yours, here's how to approach it, first make sure the roofing is safe, then cut all that top growth back which will reveal masses of twisted rooted thick stems, use loppers and small hand saw and carefully cut these away, it's long and laborious, so don't do this then decide to put new roofing up, don't beat yourself up thinking its a nightmare job do it in stages, once all this is off and you have properly killed off the main root systems then you should be ok, if you get someone in to do it then it could be expensive this is a job that you can do yourself and save money which can be used on the house, put rubbish in old ton bags to dry and burn later or take to the skip, I must say some of those buildings look rather dilapidated so be carefull.

Never thought of it having rooted into the wood. A real nightmare scenario.
Dry ivy stems do make marvellous firelighters...

Mmmm! If you take it all down the roof will most probably come with it I'm afraid :o(

The ivy and your outbuilding have joined together to become one. I see no solution other than leaving it as is in order to get as much use out of this structure as possible. Anyway from what I can see, ivy or not, advanced wood rot is setting into these old structures so squeeze out as many years of use as you can then down they go.