Is it safe to eat potatoes that have sprouted?
Not safe enough for me to try it. Potatoes actually contain a very dangerous toxin called solanine. This toxin is concentrated enough in the green parts in the plant to cause solanine poisoning. This includes the sprouts/eyes, and the potato itself if it's green.

This article from the New York Times health guide indicates that it is something to be taken seriously. Per this article, if the sprouts have been removed, and the potato is not green then it is safe to eat as far as solanine poisoning is concerned. 

However, a potato as far gone as you have described sounds disgusting. A soft potato is on its way to going bad. Where I am from, potatoes are cheap enough that it's just not worth the gross factor for me to eat a potato that has 10 cm sprouts and is squishy. I do eat potatoes that have little nub sprouts on them and that are slightly less than firm, after removing the sprouts of course.