Perhaps this is your opportunity to correct someone else's mistake. You need to show the table definition to convince us that it cannot be improved. That it may be hard work really doesn't mean it's not the right path.
This may not be possible. The data might be coming in from an external source. I imagine you've run into the old "well, _we_ don't have any problems, so it must be on your end!" scenario.
Example: we receive CSV files from an external source. These files are _supposed_ to be validated. But we have often received files where NOT NULL fields have "nothing" in them them. E.g. a customer bill which has _everything_ in it _except_ the customer number (or an invalid one such as "123{"); or missing some other vital piece of information.
In this particular case, the OP might want to do what we did in a similar case. We had way too many columns in a table. The performance was horrible. We did an analysis and, as usual, the majority of the selects were for a subset of the columns, about 15% of the total. We "split" the table into the "high use" columns table & the "low use" columns table. We then used triggers to make sure that if we added a new / deleted an old row from one table, the corresponding row in the other was created / deleted.