The key is stored in IndexedDB. You can read more about that at https://developer.chrome.com/docs/devtools/storage/indexeddb.

It's not readily available from the Terminal.app UI, but you can get there. Press Ctrl+Shift+I to open the Inspector. Select Application from the top tabs. On the left, under Storage, go to IndexedDB > indexeddb-fs > nassh-rootfs > directory. Look through the list of fullPath entries to find the appropriate entity. In the Value column, locate the data property and click the memory inspector icon to the right of it.