This is not a new section dealing with David, but only a comparison of the promise made to Aaron with that made to David, showing that the promise to Aaron was greater. Good News Translation makes this clear, although the sentence structure is a bit involved. A simpler approach is given below.

A covenant was also established with David, the son of Jesse, of the tribe of Judah may be rendered “The Lord also made a promise to [or, an agreement with] David….”

The heritage of the king is from son to son only may be translated “in which the kingship would pass from father to son” or “in which only one of his sons could become king [or, rule] after him [or, when he died].”

So the heritage of Aaron is for his descendants may be rendered “But the Lord promised Aaron that the priesthood would pass down to all of his descendants.”

An alternative model for this verse is:

• The Lord also made a promise to [or, an agreement with] David son of Jesse, from the tribe of Judah. He promised him that the kingship would pass from father to son. But he promised Aaron that the priesthood would pass down to all his male descendants.

Quoted with permission from Bullard, Roger A. and Hatton, Howard A. A Handbook on Sirach. (UBS Helps for Translators). New York: UBS, 2008. For this and other handbooks for translators see here.