Just so I'm clear on the details, this was working before the server failed? Have you checked the Event log on both instances (source and target) to see if there are any errors?
If they are making it to the target CMS, it sounds like the staging is working fine. The Web farm servers on the target should be responsible for keeping the CMS and live site in sync so it sounds like there could be an issue with the target's web farm servers. Have you checked the Web farm application to see if yours are listed there and have a Healthy status: see Monitoring web farm servers? You can also check the Tasks tab here to see if there is a build up of tasks and if there were any failures (failures should also be logged in the Event log): Web farm tasks are not being processed.
You may want to try restarting these by going to the System application when logged into the admin site and clicking the Restart all web farm servers on the General tab. If you are using Automatic mode for web farm servers, once the CMS and live site's have received a request, it will create one for each site which should then transition before hopefully reporting as Healthy after a few minutes. Once they are both healthy, it should hopefully sync these up