Double check the default browser (it may have been changed without realizing it). The only reason Internet Explorer will not open in the Start screen is if a different browser is set as the default. See the note under option one here. Even if it is already set to default, try resetting it to default again as it may not have taken the first time it was done. Try the steps listed under option one as it provides various ways of setting it as default.
The only documentation I can find from Microsoft and third party sites regarding IE opening in desktop view only always mentions a different browser being set as the default browser. I haven't found any answer besides check the default browser, so this would seem to be the only reason it should happen.