As your scenario is prolonged total anarchy, the structures that would break down first are those which need a high level of cooperation to work properly. I think one of the first victims of the breakdown would be the electric grid, as this needs a constant and careful regulation in order to work properly. Now, these days a lot of that is automated, so the grid would not fail immediately, but as individual parts of the system would no longer be maintained, it would only be a matter of time until a chain reaction makes the whole network fail. The same is true for the fuel distribution networks; although they are much less prone to local failures escalating to network-wide failure, they need much more maintenance for the components to remain in a working state. Also, a lot of it also depends on electricity.
The moment the grid fails (and it won't be gotten brought up again due to the anarchy), also the communication networks will break down. Some individual networks will continue to work for a while on batteries and/or generators, but with the distribution network for fuels no longer working, also that will fail after some time.
The same will be true for all the other networks like water, which will make the cities more or less uninhabitable, at least they will not be able to support the number of people currently living there.
Also note that modern agriculture very much depends on energy supply, as well as products like artificial fertilizer. As the various networks break down, the supply of those will break down as well.
The result will be that within at most a decade, due to the breakdown of the networks it will no longer be possible to support as many people as there are, which means a big famine with many people dying. Everyone dying will take with him the knowledge in his head; this includes a lot of knowledge that cannot be found in books. Moreover, those surviving will be full time busy with their survival, so they won't pass on most of their knowledge to the next generation either, except for those parts relevant to the current situation.
Which doesn't mean the knowledge will be completely eradicated; there will, of course, always be people who teach their knowledge to their children. But the knowledge will no longer be organized; there will be individual people with individual bits of knowledge.
Also note that today, a lot of our stored knowledge is digital, and a lot of it isn't even stored on local media, but somewhere in the cloud (think Wikipedia, or this very site we are on); everything not printed out will be lost, and even printed out material will rot, as you'd not normally would print it on high-quality material. Also printed material would be subject to rot due to no longer being properly maintained; however printed material has much better chance to survive. However books are only useful to those who can read them, and therefore the vast majority of the population will lose access to them in just one or two generations.
Additionally, in such an apocalyptic scenario, one can expect both established religions and cults gaining support, with some of the latter growing to new full-blown religions. Strong religions have rarely fared well for general people's knowledge.
However there will still be some organizations caring about knowledge, which will at the beginning be most concerned with collecting and preserving the knowledge that still exists. Which means that science will be turned back into scholarship, preserving the past knowledge instead of researching. But then, the same existed in the medieval time, with monasteries having libraries and monks being busy copying ancient texts. So that's not in contradiction of new dark ages, but actually a part of it.
Note also that for those who have the knowledge, there will be incentive to control it, as those who control knowledge control the world. Also, they will have to arrange with religion, which probably means that, just as in medieval times, the places preserving knowledge will be the same places that maintain religion, and most knowledge is restricted to those committed to the religion.
My assumption would be that the full-blown dark ages will be reached as soon as those who actually experienced the pre-breakdown period are dead.