Connect 2 external monitors to a MacBook Air (Mid 2012)
The only monitor that can daisy-chain the display connection is the Apple Thunderbolt Display. Your Asus monitors don't have Thunderbolt hubs or 'pass-through' connectivity.

However, I'm surprised that the Belkin Dock didn't work: assuming that it had a HDMI port and a TB2 pass-through, then that's exactly what it's for. 

There are TB2 to dual HDMI adaptors, which might work, if you can find one.
https://www.techpowerup.com/149981/zotac-announces-dual-hdmi-output-adaptors

If you really want to invest in 'future-proofing', you might be better served updating your Mac.