Go to [Tools, Options], section "Web Forms Designer" and enable the option "Enable Web Forms Designer". That should give you the Design and Split option again.
The solution of creating a new HTML file with HTML (Web Forms) Designer worked for that file but not for other, individual HTML files that I wanted to edit.
I did find the Open With option in the Open File dialogue and was able to select the HTML (Web Forms) Editor there. Having clicked the "Set as Default" option in that window, VS then remembered to use that editor when I opened other HTML files.