Since you e-filed your original tax return, you should be able to e-file the amended return. It's worth a try. It might get processed somewhat faster that way, though it will still take many months.
If you mail it, then yes, you staple everything together with the 1040-X and mail it. The Form 1040-X goes on top. Don't forget to sign the 1040-X. You should mail it by a service that gives you tracking and confirmation of delivery, such as certified mail with a return receipt. Keep a copy of everything that you send.
You will have to include at least the Form 1040-X and Schedule D. I would also attach a copy of the Form 2439. The Undistributed Long-Term Capital Gains will increase your AGI, so other things could be affected that depend on AGI but are not directly related to capital gains. Check to see if any other forms or schedules changed. Since you are adding income, you probably owe additional tax on Form 1040-X line 20, so enclose a check for the amount that you owe.
If your state has income tax, you probably have to amend your state tax return as well. That gets mailed separately to a different address.
