The Greek συναχθέντες κλάσατε ἄρτον καὶ εὐχαριστήσατε is not ambiguous. It means, "Having gathered together, break bread and give thanks." The verb "to give thanks" or even "to break bread" extremely early on became the noun "Thanksgiving" (Eucharist), the proper name of the celebration.
Arguably, one is more than just justified in translating the verb εὐχαριστήσατε "hold ye the Eucharist," since the significance of "give thanks" was equal to that of "break bread" in the context of the sacramental Eucharistic celebration—both are a wink to the institution of the celebration, when Jesus "took bread, gave thanks, and broke" (Luke 22:19) etc. and do not mean 'break bread' and 'give thanks' in general, but very specifically, to "do this" which Jesus commanded: to consecrate (or "bless" as St. Paul says) and receive or partake of His body and blood. (Hence the Didache and all early Christian writings on the matter calling it the θυσία—"sacrifice"—of Malachi 1 which Jesus "offered for" us, and which we offer to God.)
Κατὰ κυριακὴν κυρίου is a bit trickier, but not much, since its meaning is so well documented from other early writings.
The closest thing in the New Testament is found in Revelation, when John clearly makes reference to a day already known to his readers as "Lord's Day" (κυριακος ημερα). Given later witness to the fact that this "Lord's Day" is the first day of the week (Sunday), together with New Testament data on which day the church 'came together to break bread' (Acts 20:7), we can safely understand this to be a reference to holding the Eucharist on (κατὰ ) the Lord's [own] 'Lord's Day.'
The second closest thing is St. Paul's letter to the Corinthians.. specifically the section on the celebration of the Eucharist (1 Corinthians 11:20), funny enough. It says, "When you come therefore together into one place, it is not now to eat the Lord's [κυριακον] supper [i.e. which ought not to be]" (DRB).
But in something a bit less obscure and to the point (i.e. specifically about the day), we find the exact phrase "κατα κυριακην" as a day in a letter of Ignatius of Antioch to the Magnesians (Chapter 9):
If, therefore, those who were brought up in the ancient order of things have come to the possession of a new hope, no longer observing the Sabbath, but living in the observance of the Lord’s Day [κατα κυριακην], on which also our life has sprung up again by Him and by His death—whom some deny, by which mystery we have obtained faith, and therefore endure, that we may be found the disciples of Jesus Christ, our only Master...
And which obviously refers to the first day of the week, Sunday.
So what does "And κατα κυριακην of the Lord, [hold the Eucharist]?" mean? The κατα means either "every" or "on [the,]" while the κυριακην must refer to a day known to the readers as such and described already. Yielding not "And every Lord's Lord's," ignoring the distinction between κυριου and κυριακην but "And on the Lord's [own] day."