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Bound Noun, Josa for Nominative, Objective Case in a Complex Sentence

The josa attaches to a che-eon(pronoun, noun, susa), and thereby signifies the relation between the that word and the other words.
1.'제가 부인이 부인의 잃어버린 아이를 찾는 걸 도와 드리겠습니다.
2.'I'll help you find your lost child, ma'am.
In the Korean sentence, there are two of the same nouns(부인) performing as a subject and a che-eon modifying word.  The differences between them are having different josas(이, 의) each in comparison with the and the fact that the personal pronoun 'I' is the subject and 'you' is the object of the verb 'help', and 'your lost child' is the object of the verb 'find' in English sentence 2, without having any josa. In this case we can omit either '부인이' or '부인의', and since we have the conjugated form of '찾다' in this sentence it's understandable even if '잃어버린 in '잃어버린 아이' is left out. In addition, you can omit also '의' at times Additionally, you can sometimes omit '의'.  
Let's focus on constituents in the Korean sentence.
Firstly, we can analyze the sentence into 제가(the subject) 부인이 부인의 잃어버린 아이를 찾는 걸(the object) and 도와 드리겠습니다.(the verb phrase).
제가 is the form into which josa '가' and pronoun '저' are combined. '가' signifies '저(제)' is subject of this sentence by attaching to '저'.
The nominative case is the form which shows its relationship to other groups in a sentence is the subject. '이','가', and '께서', '에서' is are josa for the nominative case.
The example sentence is a complex sentence into which '부인이 부인의 잃어버린아이를 찾다'(Ma'am finds her lost child) and '제가 도와드리겠습니다.'(I'll help you.) are combined. 부인이 (부인의) (잃어버린) 아이를 찾는 것, the noun clause of 부인이 부인의 잃어버린 아이를 찾다, is the object of '도와 드리다'(도와드리겠습니다.) 걸 is a contraction of 것을. 것 is a bounding noun(의존명사). The contraction of josa and che-eon is stated in clause33, chapter4 of 한글맞춤법.
을, 를 are Josa for the objective case.
We call a part of clause that is not the subject, the predicate.  A noun clause in Korean is formed if a predicate in a clause or a sentence takes '-음,-ㅁ', '-기', or the bound noun '것'. For reference, a bound noun(의존명사) in Korean doesn't refer to specific words, but refers to abstract things or possessions.  Thus , in prescriptive morphology, it is regarded as a free morpheme and or full morpheme(otherwise known as lexical morpheme). '-는' as in '찾는' is 관형사형어미, which is the ending enabling the conjugated word modifying noun '것', which belongs to 체언. The josa '를' makes '(잃어버린) 아이(lost child)' perform as the object of '찾다' whose form in this sentence is '찾는' in the noun clause.

As a person studying Korean, I sometimes find using all the various particles to be overwhelming.  But as you say, it can be possible to still understand the meaning even if particles are omitted.