I just got some advances:
(1) The problem seemed to be related to the applications. Most applications which I have tested, don't seem to sensitive to the language toolbar, while a few do.
In particular, Microsoft Word and Notepad still only show alphabetic character.
(2) Open Office (actually, I have installed the variant called "LibraOffice",
didn't work initially, but after changing the options, enabling "Asian Language
Support" and RESTARTING Libra Office, I can type Hiragana.
(3) Accidentally I found, that an extension to the Chrome Webbrowser, called "Chrome Notepad", also lets me type Hiragana. For example, I can type them in Chrome Notepad and paste them into Word; Word can show them, even if it does not allow them to be input.
However, one problems remains: The input is done "as if I had a Japanese
keyboard" attached. For example, when typing the letter k, the Japanese character の ("no") appears.
On Windows 2000 (and on the Mac), I'm using an input method called "Kotoeri", which is taylored to western keyboards. For example, for writing の, you type the letters "n" and "o" in sequence.
If I could solve this problem, I would be satisfied, since I can at least work with Libra Office and Chrome Notepad, which would be sufficient for my purpose.