Thank you for uploading that .sky.zip file to the Skyline support file sharing folder.
Since the spectral library does contain ion mobility information, that means it will be easy for you to create an ion mobility database (.imsdb).
To do that, you go to the "Ion Mobility" tab at "Settings > Transition Settings".
Choose "<Add...>" in the dropdown.
In the "Edit Ion Mobility" dialog, you can push the "Import..." button and tell Skyline to import ion mobility values from the library in your document.
On the Ion Mobility tab in the Transition Settings, you also have to specify an ion mobility window width because otherwise Skyline will not use the ion mobility database.
After you have done all that you can tell Skyline to extract chromatograms again using the "Reimport" button at "Edit > Manage Results".
I am not sure whether the ion mobility database would have been created automatically if you had used the "File > Import > Peptide Search" menu item to import your MSFragger results into Skyline instead of using the Skyline document that had been created my Fragpipe.
I am not the expert on ion mobility and import peptide search in Skyline. Someone else on this support board will probably be able to give you a better answer about whether there would have been an easier way to get ion mobility filtering to happen in your Skyline document.
It certainly sounds like Fragpipe should have created you a Skyline document which used ion mobility filtering. I am not sure whether the behavior you are seeing is a bug, or whether they have not yet implemented that part of the Skyline document creation, or whether there was some other option you could have chosen in Fragpipe to get the behavior you needed.
-- Nick