Since the other two responses have addressed other aspects of your question, I will respond to your interest in Kanban. I come from a company that builds a Kanban product (SwiftKanban). We use Kanban in all our product development and we have seen the benefits of data-driven forecasting rather than estimation, which we have dropped altogether! Also, I am personally a Kanban coach/ trainer. I have worked with a number of our customers helping them with Kanban implementation, usually in software and IT. So, I am very familiar with your challenge.
If you have historical cycle time and workflow data for this project - for all the different types of work your team may be performing, you might very well be very easily able to forecast with different levels of confidence when your team will be able to complete the project, assuming the requirements are reasonably stable - and there won't be too much of a scope creep. Using your Cumulative Flow Diagram (CFD) and your cycle time data, you can either do simple linear projection to see when the remaining scope might get done; or you can do more sophisticated modeling using Monte Carlo simulation (which is what we do) or other techniques.
Using just the CFD on your project's existing data, you can easily get a projection like the one shown below for our current backlog.
Here, simply based on the historical throughput of the team, we can project when the remaining amount of work in the backlog (the grey band's vertical jump) is likely to get done.
You could also do something as basic as look at your recent throughput - and get an approximate picture of when you are likely to deliver the rest of the work.
If you use modeling using Monte Carlo simulation, you can do different analyses. For example, you can look at how much work you can get done in the next 30 days given the past performance, at different levels of probability.
Or you can define the exact amount of remaining work (number of user stories, change requests, defect fixes, etc.) and get the probability curve of the likely completion dates.
Here you see, for a given scope on the left, the distribution curve on the right for the likely completion times.
Here is a more zoomed in look at the probability distribution curve.
Depending on how you can get the historical data, you can do this in a spreadsheet or use a sophisticated Kanban software - such as ours or others in the market.
The overall premise, of course, is that the team remains stable and the work mix continues to be similar.
Hope this helps. I'd be happy to answer any questions you might have.