Welcome to the forum!
Yes, the situation you describe is exactly the kind of situation which the Paint Bucket tool was designed for. It's not a perfect solution by itself. You usually will need to "tweak" the path a little bit. In this particular case, the tweaking will be pretty simple, with Snapping. Well, or even without bothering to set up snapping, really.
Depending which version of Inkscape you installed, the Paint Bucket tool icon probably looks like this
. If you really don't have it, I guess it's possible you have a really old version of Inkscape. In the current version of Inkscape, it should be 5th from the bottom.
Please tell us which version of Inkscape you have, which package you downloaded, how you installed it, and also which operating system you're using. It's possible if it really is missing, that you didn't install the correct version, or something went wrong during installation.