Question: My title sums up my question pretty well. We're investigating whether we can use a Kanban-based system to manage and track some internal projects and one of our requirements is to track time spent on activities. This will allow us to collect stats on engineering cost. 

Is anyone doing this and if so, how?

Answer: 
Disclaimer: I am related to Eylean board development

There is a precise match solution launched instead of Eylean which covers the idea of using both Cycle time and time tracking as part of Kanban System. If it is still relevant option you can consider reviewing [Teamhood](https://teamhood.com)

One more tool is [Eylean Board](http://www.eylean.com) - it is virtual board which offer both scrum and kanban tools. You can customize it anyway you like through a simple GUI. I have to say the board looks pretty nice and is very fluent with drag and drop everywhere. And of course it has cycle and lead time reports. One thing is that Eylean must be installed on client machine it is not a web app.
I agree, that in general Kanban is not about tracking time spent on tasks. However, I understand a need for that. It seems to me that you need a virtual Kanban board with build-in time tracking and reports. The solution that offers such a powerful combination is [Kanban Tool](http://kanbantool.com/time-tracking).