Writing in Ancient Greek with babel

The perispomeni used to work. The fix is easy, though.

For the dialytika the character to use is ", not ¨.

\documentclass{article}
\usepackage[greek.polutoniko,english]{babel}
\usepackage{xpatch}

\xapptocmd{\greektext}{\edef~{\string~}}{}{}

\begin{document}

\textgreek{'a, <a, >a, `a, a|, ~a, "i}

\end{document}

On the other hand, you can also directly input the precomposed letters with the diacritics (not with the combining accents).

\documentclass{article}
\usepackage[greek.polutoniko,english]{babel}
\usepackage{xpatch}

\xapptocmd{\greektext}{\edef~{\string~}}{}{}

\begin{document}

\textgreek{'a, <a, >a, `a, a|, ~a, "i}

\textgreek{ά, ἁ, ἀ, ὰ, ᾳ, ᾶ, ϊ}

\end{document}

Here, there is my answer done with greek.polutoniko package + the \ to have the correct \~a, and \"i.

\documentclass[12pt]{article}
\usepackage[greek.polutoniko,english]{babel}

\begin{document}

\textgreek{'a, <a, >a, `a, a|, \~a, \"i}

\end{document}

I add the screenshot compiled with LuaLaTeX.

However also your code works fine if you use the correct commands.

\documentclass[12pt]{article}
\usepackage[greek, english]{babel}
\begin{document}

\textgreek{'a, <a, >a, `a, a|, \~a, \"i}

\end{document}