Even though it might seem unlikely, you absolutely can't eliminate the possibility of planes falling out of the air unless you have thoroughly audited and tested all their systems in advance. Not having planes in the air at the exact moment of rollover - just in case - might be a good idea too. Similarly financial markets and other key infrastructure.
13 years is plenty of time to make a start on all of this, and will allow the costs to be spread over that time period without undue hurry. It's also long enough that it will be possible for much software to be allowed to reach end-of-life without being audited, while applying rigorous testing to new software. Procrastination, though, is not a good option.
Why should floppy disks be a problem? Even if they are fantastically out of date, if they can be maintained as a niche legacy technology, they're not a problem. They won't suddenly stop working, it will just slowly get more and more expensive to get the parts. Supply chains are magic - at the right price, someone will always be willing to make more floppies or drives. Of course that price might be $500 per floppy and $50,000 per drive...
13 years is plenty of time to make a start on all of this, and will allow the costs to be spread over that time period without undue hurry. It's also long enough that it will be possible for much software to be allowed to reach end-of-life without being audited, while applying rigorous testing to new software. Procrastination, though, is not a good option.