With Spring15 the idea to allow last activity date to kick of workflows is marked as delivered, by process builder.
https://success.salesforce.com/ideaView?id=08730000000BqpKAAS
I think you would need to use pb to launch a headless flow when last activity date is changed. Don't think it would be possible w pb by itself.
The flow could lookup the activity where due date == last activity date (somewhat tricky given how last activity date evaluates contacts, opps, etc, but feasible) and then write the values to account.
Thinking out loud, it may be easier actually to just use pb and flow on activity as you can easily lookup all activities with a given account id, sort by due date, and write the values from the most recent activity to the account.
Either way there should be a declarative option using flow and pb.