Welcome to the forums,
I'm very sorry for your male budgie's passing and for the current predicament you are in.
Given the circumstances and the fact that you mention your female is not coping well with the loss of her mate and she has not been properly incubating the eggs, it would be best if you candled these eggs to check for fertility.
If the eggs are fertile and the developing chicks alive despite the mother's erratic ways and if she is showing signs of clutch abandonment, then I agree that it would be best to foster these eggs to another budgie pair(s) that is/are currently going through the same breeding process.
If the eggs are infertile or DIS (chicks dead in shell), then you can discard them and remove the nest box.
If you can't foster the eggs and your female decides to go back to full incubation, then you need to prepare yourself to assist her and the incoming chicks during the entire process and this means getting hand feeding supplies just in case you need to take a more active role in raising the chicks if the mother is too overwhelmed by it.
For easier access, you can place the food/water containers near the nest box's entrance. Later on, placing some spray millet inside the nest box will also be helpful.
I'm wishing for the very best outcome for your female budgie.