In "Huswifery" by Edward Taylor, how would you interpret the following words: "complete" (line 1), "near" (line 3), and "fine" (line 9)?

"Huswifery" is one of the best-known meditational poems. It was written by Edward Taylor in the late seventeenth century and serves as a means of placing the speaker in communion with Christ through metaphysical and literal means. 
Taylor himself was a pastor and physician (and Protestant dissenter) born in Sketchily, England, in 1642. He emigrated to the Massachusetts Bay Colony in Colonial America after the 1662 Act of Uniformity was established, which he refused to sign. He took poetic inspiration from the English metaphysical poets, including George Herbert, John Donne, and Richard Crashaw. Though his poems were not published until more than 200 years later—after Thomas H. Johnson discovered a 7,000 page manuscript written by Taylor in the Yale University library—he is now recognized as one of the most significant poets of his time. His body of work is categorized as both "American baroque" and metaphysical. It largely deals with his commitment to and love for God.
In the first stanza of "Huswifery," Taylor uses the imagery of a spinning wheel to convey his devotion to the Lord. He asks that God use him as His spinning wheel and that He provide to him something to hold the holy words of the Scripture. He compares his own affections to the "flyers" (in other words, the revolving arms on the wheel that twist wool into yarn), compares his soul to the spool around which the thread is wrapped, and compares his spoken words to the reel used for winding the spun wool. 
The second stanza elaborates on this metaphor as the thread is transformed into cloth. Faith becomes woven into fabric as a garment of salvation, and God's law serves as the finishing process, with the product being a beautiful cloth that is heavenly in nature.
Let's now answer your question. With its original language and spelling, the first two stanzas read as follows:

Make me, O Lord, thy Spining Wheele compleate. 
Thy Holy Worde my Distaff make for mee. 
Make mine Affections thy Swift Flyers neate 
And make my Soule thy holy Spoole to bee. 
My Conversation make to be thy Reele 
And reele the yarn thereon spun of thy Wheele. 
 
Make me thy Loome then, knit therein this Twine: 
And make thy Holy Spirit, Lord, winde quills: 
Then weave the Web thyselfe. The yarn is fine. 
Thine Ordinances make my Fulling Mills. 
Then dy the same in Heavenly Colours Choice, 
All pinkt with Varnisht Flowers of Paradise. 

In line one, "compleate" simply means "complete"—something that is whole and finished. In line three, "neate" is something that is clean and free of impurities; in other words, Taylor's affections are pure and made cleaner by being processed through the Spinning Wheel. In line nine, "fine" means something that is of superior quality and exquisite in nature. In other words, the product of this spinning is a cloth made from superb faith.