I think you have an opportunity now to set things up for the future.
You have two new staff members (one brand new, one 6 weeks in) . Use this to your advantage to put systems in place.
Let them know your expectations and why these tasks are important to you. I think you are completely within your rights to expect staff to enter this data correctly.
If it were me, I would make one staff member the office manager who is ultimately responsible for ensuring the data is correct - make this an essential part of their role. May mean supervising the work of the other staff member and checking their work (means you dont have to). If you do this, ensure both staff are aware of the heirarchy (easier to do with new staff than existing staff).
Have a meeting with the senior staff member once a week to ensure the office is running as you want it to - but take on board any suggestions made and use this time to let her know what she is doing well. If you're not good at it, make a list before the meeting. Even a quick positive email once a week can be appreciated.
You said the previous staff found client contact time stressful, so perhaps restructure the roster so there are two people on during this time. With extra time on their hands they have no excuse to get the data entry wrong, particularly with a staff member double checking. The increase in staffing costs could be offset by increased income.
Yes those are my thoughts exactly. Except that instead of me personally meeting the senior staff member and pointing out errors, I would like to outsource this to an external managment firm.
This was my problem with Mary who resigned. She was the senior staff member and the whole office and all systems revolved around her. So when she left suddenly, the whole place has collapsed. And I am trying to crawl back out of the rubble.