TechnicalDebt is a very useful concept, but it raises the question of how do you measure it? Sadly technical debt isn't like financial debt, so it's not easy to tell how far you are in hock (although we seem to have had some trouble with measuring the financial kind recently).
Here's one idea to consider. When a team completes a feature ask them to tell you how long it took them (the actual effort) and how long they think it would have taken if the system were properly clean. The difference between the two is the interest of the technical debt. (So if it actually took them 5 days but they think it would have taken them 3 days with a clean system, then you paid 2 days of effort as interest on your technical debt.)
There are certainly some serious flaws with this technique. The statement of how long it would have taken on a clean system is an estimate based on an imaginary state - so is difficult to make objective. There's the effort in capturing this information, which is easy to get out of hand. But the result may help project a picture of the state of the code-base in a way that's visible to non-technical staff.
Furthermore it may also help with decisions about whether to pay the principal. Some teams like to add technical debt stories to their product backlog - with estimates on how long it would take to remove them. Such technical debt stories are also estimates, but also provide a picture of how much debt has built up. You could get a bit more clever with the estimated interest payments by apportioning them to these debt stories (I spent an extra day on this feature because of the bad state of the flipper module). Comparing interest payments with the principal may help inform a decision about whether to pay off the principal.
I ran into someone recently who tried something a little like this and found it handy, but it's not something I've run into a lot. Certainly there are flaws with doing it - but it may be worth a try for a few iterations.
Update: A recent discussion surfaced another way to capture the estimated interest. During a retrospective (which wise teams do at the end of each iteration) capture an estimate of interest paid against each of the problem areas of the system. Doing this estimate against recent completed work may be easier than forward estimates against future stories.