The proper way to file Form 8606 is attached to Form 1040-X.
Form 8606 can be mailed by itself only when you are otherwise not required to file a tax return.
Do you have the correct taxable amount on Line 4b?
Get Form 1040-X from IRS website and mail it in with your 8606, which you can also get in fillable PDF.
Note: since you are not changing any dollar amounts on your amended tax return, you can leave all the lines 1-23 EMPTY.
For an inherited IRA, the names on 8606 should be "<owner> and <your name> BENE" or Beneficiary if it fits, and your SSN.
Part III explanation: "didn't include Form 8606 with e-File".
You will have to mail it so this does not use up your one 1040-X e-File.
