It's hard to say what her contribution or influence has been on music. Or actually, it's not that hard. First, you must explain that Madonna would still not be present today as a musical artist without the music being good. Sometimes damn good and sometimes outright brilliant. If her acting career had taken off, then we could say she's remained in the public eye for that reason or any other of the careers she attempted; author, mogul, etc. But, it's always been the music.
Like the review of Madonna said, she brought credibility to dance music. If you think of or listen to music of the 80's, she's one of the few artists that was not a victim of the time. Her music was always ahead of the curve in sound and in production. Thus, her music has been timeless. And it continued to be in the 90's. She was probably more influential musically speaking in the late 90's, early 00's. After "Ray of Light", every other pop song released had an electric sound to it. A blip here and a bleep there now constituted what a pop song should sound like. She added an element to the structure of pop music affected by a single release of work. (Of course, she was influenced as well, but U2 and David Bowie couldn't master that influence, Madonna did).
Now, what the Beatles are to albums and "I Love Lucy" is to television, Madonna is to live concerts. She simply revolutionized live performance. In the process however, she never lost one major element: connection with the audience. And though some will say that the production is to cover her least greatest attribute, her voice, some of Madonna's greatest performances are of her alone singing to the audience. People will come to realize later that the production only added value to one of the greatest artists that ever lived. I believe she is, and her live shows prove that. She isn't just a rock and roll singer, she is a musical artist. An artist period.
Lastly, speaking of artistry, all great artists are said to have had a lasting cultural impression. Madonna has already left an amazing track record of cultural influence. She was a sex symbol and overt sexual artist during the Reagan 1980's. A post-AIDS commenter throught visual and musical art. Her views on homosexuality in print, music, and music video are absolutely rare in the history of rock and roll. The last twenty years saw the biggest change in acceptance towards homosexuality. No doubt her influence contributed to that. And now, she is having the last laugh as a rare woman artist to remain relevant past her 40's and now into her 50's . She's affected cultural mores and ideals by being a woman of sexual independence (while retaining a feminist quality) and by waging a war against agism. Maybe there's more that I missed or too much that I've written already, but there it is.
And of course, she affected music visually by being, perhaps, the greatest video artist ever. And don't let anyone give you shit for that. The Beatles were the first to make music videos. And they did it so they didn't have to perform live. Madonna does it (not so well anymore - proper music videos anyway) and still performs live, where we still see visual video versions of her music.