@Daniel Goldwater If you have your notebook setup under your user directory the files that are created and saved should be available in your default storage account i.e the storage account connected to your workspace under the fileshares of your storage account. For example, here is a screen shot of my notebooks under my username from the studio.
By default, all these files are mounted to your compute instance when you start the instance, and this enables you to connect to git through terminal and use an external source control mechanism. Since you have not added this the files that are added previously should be available here even if you have deleted your compute. The files should be visible under your user account even when no compute is available.
The file share that is mounted is from your storage account as seen in the screen shot below:
This file share should be visible under your storage account file share tab and you should be able to download the data from storage explorer till the point the data was last saved. Ideally, notebook data is saved every 30seconds or whenever they are saved or when a checkpoint is created. Here is a screen shot of my storage account from storage browser to help you check the same on your account.
I hope this is helpful!! Thanks!!