Step 1: Download
Download the latest stable versions of Apache and PHP.
Step2: New Apache
Unzip the Apache download into [laragon installation dir]\bin\apache
Here it looks like: c:\Development\laragon\bin\apache\apache-2.4.20_64vc14
Remember to copy the httpd-vhosts.conf (virtual host file) from the working Apache.
I also changed the content of the httpd.conf (so it points to the correct directory). Perhaps Laragon does this automatically, IDK.
Step 3: New PHP
Unzip the PHP download into [laragon installation dir]\bin\php
Here it looks like: c:\Development\laragon\bin\php\php-7.0.6-64-ts
Change the php.ini to have the same extensions as the working php version.
(you can use a visual compare tool, like Winmerge, or Meld)
If you use xdebug, beware that a newer version of php also uses a different version of the xdebug setup.
Step 4: Try it
Now you can start Laragon, select the new 2 things and that should work.
Perhaps I forgot a few steps, but if you try and not succeed, please let me know, so we can figure this out.