All the documents you require are listed clearly on the application form. It's much easier than your previous visa with less documents required. ILR is rarely refused, unless you don't qualify for it (i.e. you haven't taken the Life in the UK test or have not reached 2 years in the UK before you apply).

The basics:

• your passport and passport photo
• your spouse's passport (or full photocopy of) and passport photo
• Life in the UK test pass certificate
• your marriage certificate
• your last 6 months of bank statements and payslips (from one or both of you, depending on who is earning) and any savings account statements
• any evidence of benefits if the UK citizen is claiming any (the US citizen cannot have claimed any)
• 6 letters addressed to both of you at the same address, spread evenly over the 2 years. Either 6 jointly addressed, one per 4-month period, or 12 individually addressed, 2 per 4-month period, or a combination thereof.

The applicant must also have no unspent criminal convictions and must have passed the Life in the UK test before applying.