"Umlaut" works -- except for Ü

As several have noted, the font has problems. Not only the missing umlauted characters, but also a capital U with zero apparent height.

If the use of the font is necessary, I suggest fudging the umlauts using a tailored version of \umlaut, originally provided in Looking for a package/method that can make "best-guess" (heuristic) accents onto a font that doesn't have latin-extended support.

Normally, \umlaut U would be sufficient, but because of the bad height of the U, \umlaut{U\vphantom{A}} is required in this case.

\documentclass[11pt,a4paper]{letter} % Specify the font size (10pt, 11pt and 12pt) and paper size (letterpaper, a4paper, etc)
\usepackage[utf8]{inputenc}
\usepackage[english, german]{babel}
\usepackage{hyperref}
\usepackage{graphicx} % Required for including pictures
\usepackage{microtype} % Improves typography
\usepackage{gfsdidot} % Use the GFS Didot font: 
%http://www.tug.dk/FontCatalogue/gfsdidot/
\usepackage{miama}
\usepackage[T1]{fontenc} % Required for accented characters
\usepackage[official]{eurosym}
\usepackage{setspace}

\usepackage{stackengine}
\newcommand\fit[3][.3ex]{\stackengine{#1}{#3}{#2}{O}{c}{F}{T}{S}}
\newcommand\umlaut[1]{\fit[.05ex]{\scriptsize..}{#1}}
\begin{document}
\begin{letter}
{Address}
\opening{
Sehr geehrte Damen und Herren,}
blah blah\\
a \umlaut A ä\\
o Ö ö\\
u \umlaut{U\vphantom{A}} ü\\
\closing{Mit freundlichen Grüßen \LARGE \fmmfamily  }
\end{letter}
\end{document}

If compiling with LuaLaTeX or XeLaTeX instead of pdfLaTeX is an option, consider using the OpenType font (packaged with gfsdidot as well) instead of the Type 1 font which appears to be buggy:

\documentclass[11pt,a4paper]{letter}

\usepackage[english, german]{babel}
\usepackage{graphicx}
\usepackage{microtype}
\usepackage{fontspec}
\setmainfont{GFS Didot}
\usepackage{miama}
\usepackage[official]{eurosym}
\usepackage{setspace}
\usepackage{hyperref}

\begin{document}
\begin{letter}
{Address}
\opening{
Sehr geehrte Damen und Herren,}
blah blah
a Ä ä\\
o Ö ö\\
u Ü ü\\
\closing{Mit freundlichen Grüßen \LARGE \fmmfamily  }
\end{letter}
\end{document}