There aren't any Kanban-specific rules for how to run a standup. The concept of a regular - often daily - synchronization meeting exists in a number of methodologies and frameworks, and the same techniques can be applied to a team using Kanban.
I do have three recommendations for improving the standups:
First, don't go "around the table". Walking the board techniques, in my experience, tend to be more effective. Start with the work that is furthest "right" - or closest to being done - on the board. Figure out how the team can get those items done. Then, work backwards, figuring out the best way for the team to progress those items forward and identify anything blocking progress.
Second, find a way to stay out of the messy details. Chances are, even if people are interested in those details, they aren't going to be involved. Bring up the issues, identify who needs to be involved in getting the work progressed closer to done, and coordinate getting the people together and problems solved. Get together and solve the problem outside of the standup.
Third, find a way to get to one standup a day. Even if it's at the end of one day for one team, that team can plan their next day. Reducing the number of meetings like this will reduce context switches in the day and give the team a little more focus time to get work to done.