The sick person got the important information in some way. What is the cost to have some other person get the important information again without help from the sick person? This cost can be money or time or some other resource.
You played poker by having only one person with this important information and fate called your bluff. Pay the price, move on, do better next time.
You could ask the employee via SMS that you need exactly the important information XY and if they see a way to give it to you or if it is better to start for with another employee from scratch. This will show the sick person how important the information is, but also that they are not pressured into giving the information if they are just not able to.
In one of the companies I worked there was a huge wiki where really all information of every project went. Something was missing and you found the person who had it? You let them write it down or wrote it down yourself, but the next time someone needed the information, he was able to search for and find it. I think a wiki is more powerful than a bunch of files on a shared drive, because searching for stuff is a lot faster.
If the important information is a password or login information, make sure you have a company wide password manager, that your IT staff knows how to access emails, for example when you click on a forgotten password link. The email used to access other services should also be documented in the wiki and/or password mananger, and so on.
Create an enviroment where no data is stored in the heads of people but everything on real storage media, accessible by all, or at least with access grantable to everyone by your IT staff.