You star. I've trying to work out which Windows keyboard mapping is best yo use for this. It's a bit clunky on a UK English physical keyboard, as we have the @ sign on the physical key that combines for the umlaut, but it's still easier than remembering the Unicode codes.
For anyone else wondering how to do this, in Windows 7 add ''English (United States-International)'' as an input language, and activate it from the language bar. Then You need an umlauted character, hit the double-quote '' (if you have a US keyboard) or @ (if you have a UK keyboard), release it then type the letter you want the umlaut on. So, '' then 'a' gets you ä, '' then 'U' gets you Ü, and so on. For eszet ß, use AltGr-S.
Once you have finished your Duolingo session, you can go back to you use usual keyboard by choosing that in the language bar.