This is a deeply-nested sentence, but the "outer" structure is:
［この羽織は、(long relative clause here) ものだ］と、母の手紙に長い説明がある。
In Mom's letter, there is a long explanation that says "This haori is (such-and-such) a thing."
と in bold is quotative, and everything before it is a content of the explanation. In the quote, この羽織 is the subject, and its corresponding predicate is ものだ which is modified by a fairly long relative clause.
Next, let's consider the "inner" part, the long relative clause modifying もの:
This roughly translates to "The thing (haori) which was woven and dyed with a kamon by Omitsu's mother in Miwada and then sawed and finished by Omitsu (herself)".
の in bold is interchangeable with もの ("thing"), and in this context it refers to the new fabric before dyeing. This の is working as the object of the following two verbs, 染める and 縫い上げる.
You may be unsure about how the large relative clause modifying もの works because there are already subjects and objects in it. If that is the case, consider the following simplified "double-subject" base sentence:
Regarding this haori, Omitsu's mother dyed cloth, and Omitsu sawed and finished it.
Which can be relativised to:
the thing made by Omitsu's mother dying cloth and Omitsu sewing and finishing
Similar exmaples are found here.