Your question reminds me of someone in my life who often asks highly specific (and odd) questions that turn out to be step 27 of some plan to solve something. She races through the plan until she hits a roadblock or lack of knowledge, and then asks just about that roadblock, without backing up to see if there is another way forward.
In your case you have this chain
- someone is underperforming
- you have speculated assorted possible reasons for the underperformance
- one of those reasons you think you could fix, thus being able to keep the employee
- you've tried asking the sorts of questions that would often reveal this to be the reason, but haven't got a clear answer
- so you ask how to get a clear answer
Some people are happy sharing their private lives with their managers. Some aren't, even when sharing personal problems would result in the manager "giving them some slack" around poor performance. I have had poor performers who I am sure have something in their life that is causing it but who will not discuss it. That's their choice.
Set up a one on one with this person and let them know it's to discuss performance. Lay out, quietly and calmly, the specific problems you have - task A took two weeks when we all thought it would be a few days, and more importantly you didn't tell anyone you were struggling with it; task B had to be reassigned to someone else because you couldn't finish it; etc. Then pause. Ask if the person understands and agrees that at the moment their performance is not where it should be. If they don't agree you can lay out more examples, or you can say that it's not up to them to make this judgement and that as their manager you are telling them they are not performing at the level they need to be.
This is an emotional moment. They are wondering whether you're about to fire them, demote them, take their best project away, or what. Do not leave them hanging out there. If you are not going to do anything right now, say so clearly and simply. Something like "I'm not going to change anything about your job right now. I just needed to make sure that you know that right now, your performance is not good enough and needs to improve." Make sure they have that reassurance.
Then say "sometimes, good performers hit a rough patch because there's something else going on in their lives. We are a supportive company and if you're facing something hard, and just need some time to get back to your usual self, just let me know." Then wait. And wait. They may say there's nothing. You may not believe them. So be it. Give them the dignity of denying if they want to. They're going to take the consequences, and they are an adult.
If they now tell you about a problem, you are obliged to put up with poor performance since you said that you are that kind of supportive happy company. In my experience you get rewarded for this with great loyalty later. So reassure them again and end the meeting with something about looking forward to them being back to their old self, and reminding them that if they need time off to deal with things, just ask. According to your question, this is what you wanted.
In the absence of an admission, lay out for them what you want and when you want it by. "I need your speed to get back to what it was 6 months ago, and for you to complete tasks within your estimates. I also need instant notification when you're slipping from your estimates so we can adjust plans to deal with that. I'll be monitoring your work items, checkins and timesheets and I'm going to meet with you once a week to let you know if you're improving and if your improvement is enough. It's only fair for me to tell you that although I'm not taking any action today, if you can't improve this will end with you leaving us. I'm hoping it doesn't get to that point because you've contributed a lot here and at your usual pace you're a valuable member of the team. I'm looking forward to seeing your usual pace return."
Then one more chance for questions or "anything else I should know that might be relevant to all this? Anything at all?" and you end the meeting.