Accents on Immersion
I find that when I am translating sentences in "Immersion" the keyboard showing accents doesn't show up. This is probably the design and I think that it would be easier to keep the keyboard you find in normal lessons because I have to copy them from a website now!
Depending on your computer set-up you can type accents directly using key strokes. Typically on Windows platforms these are accessed by pressing and maintaining pressure on "Alt" then the numbers as follows. à = 133, è = 138, ì = 141, ò = 149 ù = 151 etc. Hope this helps. If you don't have a typical set-up try accessing the keyboard control panel of your computer to find how to do this.
This is one way you can do it and it's the most complicated one. An easier way, if you're running windows, is to setup the US-International keyboard layout. Accented letters become easily accessible by typing a ', ", ^ or ~ before the letter you want to accent (many are also accessible by two-key combination such as right alt+a making á)
The US-international layout is also available in Linux (in Ubuntu it's under Keyboard in the system settings). But I find it quicker just to use the keyboard layout of whichever language I'm typing in. In Ubuntu you can also set up a hotkey (I use alt + caps lock) to quickly cycle between your favourite layouts. It's very convenient to be able to type accented characters with just one keypress each.
Are you going to use the language you learn only on Duolingo? You'll need to learn to type all those accents without any virtual keyboards if you want to write emails or chat with native speakers elsewhere. I'd say that little keyboard with accented letters is mostly for beginners who are just trying Duolingo and don't want to bother with keyboard layouts yet.
I second the suggestion about US International keyboard layout. It is the easiest way and it does not make typing in English harder.
Also, if you happen to be using a reasonably recent version of OS X, they've added a nice feature where you hold down a letter and a little bubble pops up with all the various accents for that letter (because who really needs to hold down a letter to type it repeatedly). You can click on the one you want, use the arrow keys, or press a number key to select it. Unfortunately it doesn't work in Firefox (my default browser) so I've started using Safari for Duolingo. It probably works in Chrome.