Find the correct verb and subject

How do you break the sentence into verb and subject. Please consider me an idiot.

Samichange did a pretty good treatment above. But, let me try to break it down further for you.

Noun: Anything that is a name of a person, place or thing
Verb: Action (run, walk, climb, sing etc)

Any grammatically correct sentence needs to have atleast 1 subject-verb pair. Subject is the 'doer' of the action.

Example: I like to sing. Subject : I (the doer ) and like (verb). For determining the subject of a sentence, ask the question: Who is doing the 'action'? In this case, 'I' is the doer of the action 'like'.

Usually, the sentences are of the form, [subject] [verb] [object]. Dont worry about the object part as anything that remains after removing the subject and the verb becomes the object. Object is anything on which the'action' of the verb is acted upon.

Treating your sentence by a similar logic,

My gift for walking and talking simultaneously did not fail to attract attention

first of all, what is the 'action' here? "did not fail". Thus this becomes the verb.
Who is the 'does' of the action? "My gift", thus this becomes the subject.
"to attract attention": this is the remaining part of the sentence after you remove the subject+modifiers (acting as adjective here!) +verb . Thus "to attract attention" becomes the object of the sentence.

"for walking and talking simultaneously" is a prepositional modifier that adds more information about the subject , My gift (by saying that 'my gift' is not just any ordinary gift but it is a gift for walking and talking as well).

This is how you break down a sentence into the subject and the verb.