I'm not an expert on calculating the taxable portion of an ESA distribution, but the general rule for all income tax purposes is the IRS doesn't have to award any adjustment that you can't prove.
“Deductions are a matter of legislative grace;
taxpayers bear the burden of substantiating their
claimed deductions by keeping and producing
records sufficient to enable the Commissioner to
determine the correct tax liability.” INDOPCO,
Inc. v. Comm’r, 503 U.S. 79, 84 (1992)
If you can't prove any basis, and you are audited, the IRS is allowed to assume the worst--your basis is zero and the entire excess amount is taxable.
One position you could take is to guess at the amount of prior contributions and hope you aren't audited. Another position would be to assume you have zero basis (since you can't prove a basis), and pay tax on the entire excess distribution. I can't give you specific advice on what you should do. You may want to consult a tax professional.
