Answer:
"of my visit" - Adjective phrase modifying "object"
"of his earnest desire to see me" - Adjective phrase modifying "spoke"
"of the solace" - Adjective phrase modifying "expected me to afford him"
"that he spoke" - Adverb phrase modifying "was"
"to see me" - Adverb phrase modifying "desire"
"of the solace he expected me to afford him" - Adverb phrase modifying "spoke"