It hasn't been updated for a while and I'm not using it currently, thus can't confirm it is still working as desired (the download statistics suggest it being used though), but given there's not much to it, you should be able to achieve the first part of your use case with the Bamboo Notifier, which allows you to Trigger a Bamboo build upon successful completion of a Jenkins job.
The second part should be covered by the Bamboo TestNG Parser task, though you'll need to push your existing test result files to Bamboo by some means of course, possibly by Using the SCP task in Bamboo.