“eye and heart are at a mortal war”
The whole sonnet is built on an extended legal allegory: the poet's eye and heart are adversaries in a courtroom trial, each claiming the right to 'possess' the beloved. 'Mortal war' means both a deadly, bitter conflict and a lawsuit between mortals. The conceit runs through every line, from the plea and defendant (635-637) to the empanelled jury and verdict (639-644).
historical The sonnet form“divide the conquest of thy sight”
'Conquest' here means something won or seized by battle — the prize both faculties are fighting over. 'Thy sight' is ambiguous: it can mean the act of seeing you, or the image of you as a visual object. The eye wants to possess that image; the heart claims you already dwell within it.
“closet never pierc'd with crystal eyes”
A 'closet' is a private inner chamber — a small, locked room used for private prayer, reading, or keeping valuables. The heart imagines itself as such a chamber where the beloved lives, sealed against the eye's intrusion. 'Crystal eyes' are clear, transparent eyes (a conventional Elizabethan compliment), which the heart claims cannot see inside it.
“impannelled / A quest of thoughts, all tenants to the heart”
Both words are legal terms. 'Impannelled' means formally selected and sworn in as a jury (entered on the 'panel,' the official list of jurors). A 'quest' is a legal inquiry or inquest — a jury summoned to determine a question of fact. The jury here is made up of the poet's own thoughts, who are 'tenants to the heart': they live under the heart's jurisdiction and so are hardly impartial.
historical The sonnet form“clear eye's moiety, and the dear heart's part”
A 'moiety' is a legal term for a share or portion — strictly, one of two equal halves, but used loosely for any assigned portion. The verdict gives each faculty its proper 'moiety': the eye gets the beloved's outward, visible form; the heart keeps the inward love that cannot be seen. The couplet makes the split explicit: 'mine eye's due is thy outward part, / And my heart's right, thy inward love of heart.'