I can't speak for AmericanMother, but some of us have had to be trained in all these protocols and ceremonies.
AmericanMother is quite correct about the passage - it is read at every Service of the Most Noble Order of the Garter. It is considered to sum up the noble ideals of Knightly Chivalry that lie at the core of what that Order is meant to stand for.
There is no requirement that it be used at a funeral of a Garter Knight or Lady Companion, but it is a traditional choice, and Lady Thatcher was, rightly, proud of her elevation to the highest of England's Orders of Chivalry.
The Most Noble Order of the Garter, founded in 1348, is the highest order of chivalry existing in England and is dedicated to the image and arms of St. George as England's patron saint. It is bestowed on recipients from British and other Commonwealth realms. After peerages (and after the Victoria Cross and George Cross), it is the pinnacle of the honours system in the United Kingdom. Membership in the order is limited to the Sovereign, the Prince of Wales, and no more than twenty-four members, or Companions.
The Garter is the pinnacle, from the point of view of chivalry.