At this point you just have to make the best guesses you can and support them. Your problem is compounded by the special reporting required when stock acquired via an ESPP is sold, i.e., the need to determine if the sale of the stock creates reportable compensation, which gets added to the basis of the stock sold.
If the cycle of stock purchases was regular, e.g., money was deducted from paychecks throughout a calendar quarter with the purchase occurring at quarter end, and assuming the stock was publicly traded, then you could use the stock price at each quarter's end as a reasonable approximation and apply the ESPP discount to that. In the unlikely event the IRS ever questioned the basis used you'd bring out your supporting worksheets showing how you developed your figures.
