"...because our toddlers lost the physical remote."
How come everyone always blames their kids when this occurs? For some unexplained reason, no adult here has ever lost a Roku remote.
Generally, the easiest method is to try purchasing a universal streaming remote compatible with RokuTVs. (Some $10-15 ones available at Walmart, Target, etc).
Otherwise, if you can remember what your old SSID (network name) and password was for your old wifi connection (how you used to connect to network from previous home), you can enter/setup those credentials into your new router settings and should be able to reconnect. (The Roku devices remember the last known network it was connected to.) Same procedure can be used to connect to a mobile hotspot - just use your old credentials when naming the hotspot.
If the "Recovery Mode" of your RokuTV is actually equivalent to or regarded as a Factory Reset, then this procedure will not work, and unfortunately, a physical compatible RokuTV remote will be needed to complete setup or proceed through the Recovery Mode. (The remote will use IR commands to communicate with TV).
If your RokuTV has started to initiate Recovery Mode, then likely a remote will be needed. If only the Recovery Mode option has been presented, but not initiated, possibility that you can use the above procedure and then use the mobile app remote again to exit this option.
Roku Community Streaming Expert
Just another Roku user... I am not a Roku employee.
Insignia RokuTV, Ultra 4660, Premiere+ 3921, Express 4k+ 3941, Streambar 9102