I very well remember my son Rychard who had asthma being exactly like a limpet when he was a wee toddler, and it was exhausting, really knackering.
He was on a nebuliser which he hated so he was clingy beyond words. What we did as best we could was to prop him up on pillows in whichever room we had to do anything in and spend 10 minutes or so with a little sort of ritual cuddle, then begin to tell them a story. As the story progressed I'd begin to do whatever I needed to do in the room, tidying up whatever and then we'd move onto the next room and repeat the situation, substituting songs etc when I ran out of stories. It's a way of still paying attention to the child and indulging it's need for comfort whilst not actually having it clinging to you and eventually I found he took to it quite well. I then lengthened the time involved that I was physically away from him, and always came back and gave him a hug or cuddle after, so he knwe he'd not lost the comforting contact for good. It did take a while though and this lasted through for years as well, so crete and mnko are absolutely right, get some help from your health visitor if you can and get a break, but this does at least enable you not to be sitting down all day with them clinging to you, although they are bound to object at first.
Of course the more they cry and get upset the worse the asthma is, so they feel worse and cry more and make it worse again, it's a viscious circle, so I really do feel for you.