I note that many versions (eg J P Green, NRSV, NASB, ESV, etc) agree with the KJV and have "king and his redeemer". However, some others (eg, NIV, Complete Jewish Bible, NAB) have "king and redeemer". Clearly, there is debate about the validity of the "his" upon which I will not comment here.
Let us assume that that the first version (as per KJV et al) is valid. The traditional answer is to assume this to be a reference to Jesus. Compare Isa 43:3, 11, 45 17, 21 with Matt 1:21, Acts 4:12, 2 Tim 1:10, Titus 2;13, 2 peter 1:1, 11, etc. That is, Jehovah/YHWH claims to be the one and only Redeemer and Saviour, and, the NT asserts that the only Saviour is Jesus. Luke 1:68, 24:21, Matt 20:28, Mark 10:45, 1 Tim 2:6, Heb 9:15, 1 Cor 6:20, 7:23, Gal 3:13, 4:5, Titus 2:14, 1 Peter 1:18.