You can find a YouTube playlist here with many of the songs suggested below.
Gathering
The opening of the psalm and singing of the greatness of God is probably the strongest theme for gathering in the readings with many well known songs to choose from. Classic hymns to consider would be How great thou art (CH 154 / MP 506), Let all the world in every corner sing (CH 122) and Praise to the Lord, the Almighty, the King of creation (CH 124) while Great are you Lord (CCLI), Build my life (CCLI) or 10,000 reasons (Bless the Lord) (CCLI) are some possible worship songs. Some less well known songs that are worth exploring are the French song Praise, I will praise you, Lord / Je louerai l’Éternel (CH 175), which can be led by anything from an organ to a band, and Come you thankful people come (Resound), which has a simple call and response style making it very approachable for all ages.
Word
Great is the Lord (MP 199 / CCLI) is a fairly old worship song now but is worth revisiting for today’s psalm. For a metrical setting God is great in Zion (Grace) sung to St Gertrude and Great is the Lord our God (PFAS 48A / Hymnary) sung to Diademata are both worth considering and the choice of tune for each really suits the text of the psalm. Although not a paraphrase of this psalm, it’s also worth mentioning the famous hymn by Isaac Watts and Robert Lowry, Come, we that love the Lord (Marching to Zion) (Hymnary).
Response
Go heal the sick (WGRG) is based on Jesus’ commission to his disciples in the Gospel to heal the sick and cast out demons. God sends us forth (GIA) and Come with me, come wander / Sing hey for the carpenter (WGRG / STF 462) are more general songs about Jesus’ call and the latter also links to people in his hometown recognising him as a carpenter. We can also focus on how the call to mission is also a call to live simply in When the hungry who have nothing (CH 258) and Simple living (MP 1251 / CCLI / Townend)
The last few verses of the Epistle give us strong themes to respond to. Firstly, reliance on the grace of God, which of course will lead many to think of Amazing grace (CH 555 / MP 31) or it’s popular Chris Tomlin version Amazing Grace (My Chains Are Gone) (MP 1151 / CCLI). Your grace is enough (MP 1383 / CCLI) and Yet not I but through Christ in me (CCLI) are other worship songs worth considering for this. There is also the theme of power made perfect in weakness which are the focus of Give thanks with a grateful heart (CH 180 / MP 170) and the gospel song Just a closer walk with thee (Hymnary).
There are few songs which explicitly refer to King David or his anointing, which is ironic given that he wrote so many himself, so songs with a theme of God being with the nation are probably the best fit. God is our strength and refuge (MP 188 / Jubilate) takes words from Psalm 46 which links with the city of God, while God of this city (CCLI) and Your hand, O God, has guided (CH 511 / MP 705) fit with the broader theme.
Sending
The sending out of the disciples in the Gospel is a natural way to send out from the service and there are many songs which can pick up this theme in a range of styles from classic hymns such as Go to the world! Go into all the earth (CH 683), We have a gospel to proclaim (CH 363 / MP 728) and For my sake and the gospel’s, go (CH 248), more recent worship songs like Hear the call of the kingdom (MP 1282 / CCLI / Getty) and Mission’s Flame (CCLI), or songs from around the world such as Sent by the Lord am I (CH 250) from Central America and Murassala (WGRG / H 82) from South Sudan which calls on us all to be ambassadors of Jesus.