Try connecting the MBAir directly to your router using an Ethernet cable and a Thuderbolt to Ethernet Adapter to bypass any WiFi issues which can be difficult to troubleshoot. A direct connection should be faster and more reliable.
Boot into Recovery Mode (Command + R) or Internet Recovery Mode (Command + Option + R). If you "deleted" everything, it might be best to erase the physical drive using Disk Utility just to be sure everything is clean. You may need to click on "View" in Disk Utility and select "Show all devices" in order for the physical drive to appear in the left pane of Disk Utility. Erase the drive using GUID partition and MacOS Exteneded (Journaled). Quit Disk Utility and select the "Reinstall macOS" option.