If you're already moderately in shape (i.e. able to do some proper hikes of 20km with 10kg weight as you say) I don't think you'll have to to any specific training per se... But of course training always helps:
- For my first longer treks (200-300km, 2-3 weeks) I did some three to four 20km training hikes with a 15-20 kg pack each.
- For someone unaccustomed to hiking (not in your case), I'd suggest starting with 15-20km hikes without weight, then gradually increase it until max pack weight is achieved (maybe over a total of 10 hikes).
This really mostly serves to make sure your joints and sinews are used to being used and can bear the load - you'll likely not build up muscles this way unless you do a lot more training.
Note: Carrying 20kg for hours is not pleasant at first: expect some pain, some soreness, etc. Over the first couple of days the unpleasantness normally recedes (also your pack'll get lighter as you consume food and fuel). Until then I suggest playing around with all the fitting options of your pack: tightening/loosening some straps, moving the weight closer/further away from your back, changing distribution of weight between shoulders and hip, etc.*
Just as important as training your body is making sure that you have the right gear, which fits you properly and is comfortable on long 20kg-pack treks. Most important:
Make sure you have proper hiking boots, which fit you well, and which are well worn in. If you expect to carry a load of 20kg I'd also suggest using higher/heavier boots which also support your ankles, and which often have a stiffer, sturdier sole. (Some people trek with super light footwear, but unless you're quite experienced doing so I'd suggest wearing 'old-school' proper hiking boots.)
If you get blisters, deal with the issue before your trek: changing footwear, hardening your feet by doing more training hikes, or bringing blister patches can all work - this depends on your preference, your means, your resources, etc.
The backpack needs to fit well and needs to be properly fitted to your body. I suggest doing some training hikes with full weight (just pack in 20kgs of weigt, e.g. water bottles) to see if the backpack works for you with heavier loads. If your new backpack gets damaged at all during those hikes bring it back to the store and demand a refund for the crappy gear they sold you (happened to a friend twice). On a 100km+ hike you must be able to depend on you pack: failure is not an option, so it must be able to take some punishment.
I suggest to take advantage of your training hikes to familiarize yourself with all of your important gear and to make sure everything works the way you planned it to. E.g. cooking on your stove, putting up/taking down your tent, organizing your pack, etc.
Also the training hikes will let you get a first experience on how much food, fuel, water and other necessities (toilet paper, soap, sun blocker, ...) you'll need - allowing you to only bring to your final trek what you'll actually need.
Fun fact: the route you linkd in the question above was actually part of my first proper trek. In 2006 I trekked the Kungsleden from north to south (Abisko to Ammernäs).