Miranda Rights - Access to Attorney

The 4th point of the Miranda Rights states: "If you cannot afford an attorney, one will be appointed for you before any questioning if you wish. Do you understand?." Who determines if you cannot afford the attorney? My friend was denied an attorney because he owned a house - but he has no access to the co-owner of the house in order to obtain a lein (that she would not grant him permission to do anyway - as they are separated). He had no liquid assets. Does he still have rights to a public defender?