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“If that is true, how do we know when our works are complete to assure salvation?”

“When you pass from this life to the next.”

The conclusion can only be that there can be no assurance of salvation in this life.

There is no peace with God because we cannot know this side of the grave if He is satisfied with the payment made for our sins - original and subsequent.

There is, therefore, no Gospel.

Contrariwise, by trusting in the full payment for all sins by Christ, we can say with Paul, Romans 8 that there is no condemnation of those who are in Christ.

The conclusion can only be that there can be no assurance of salvation in this life.

That's not correct. We are assured of salvation by Christ's sacrifice, we don't worry if we've "done enough", because we keep doing positive actions under God's grace.

To put Church beliefs succintly:

Jesus alone has saved us. Jesus has saved us by his life, suffering, death, resurrection and ascension of Jesus Christ. None of us earns salvation. Redemption is a gift from God. Good works flow from the grace of God which abides within us. Good works flow from our cooperation with grace Works demonstrate the grace of God within us. Grace empowers us to be good and to do good. All the good works in the course of our lives, however, cannot earn us salvation.

five hundred years ago, and now, no Catholic authority would say that "by good works we can work our way into heaven." We don't earn salvation; God has gifted us with salvation

Also, Lutherans do not say that "good works are not necessary as part of the Christian life." In fact, as per the Declaration on the Doctrine of Justification of 1999, both Catholic and Lutherans agreed on a common position regarding justification.

The Council of Trent states: "We are said to be justified by grace because nothing that precedes justification, whether faith or works, merits the grace of justification. For 'if it is by grace, it is no longer by works; otherwise,' as the apostle says, 'grace is no more grace' [Rom. 11:6]"

We receive initial justification purely by the grace of God, not by any deeds or merits of our own.

And Paul talks about us not boasting of our precedence (like the Pharisees) but saying that "For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand, that we should walk in them" (Eph. 2:10). God has raised up both of us-Jews and Gentiles-to sit in the heavenly places in Christ Jesus, for we received initial salvation as a gift. We obtained it by faith in Christ (which itself is a gift from God), not by works of obedience to the Mosaic Law. So neither Jew or Gentile can boast over the other of having privilege with God

Remember that the only instance of the phrase sola fide is in James 2:24 where sola fide is rejected.

The Catholic Church teaches only Christ is capable of meriting in the strict sense—mere man cannot (Catechism of the Catholic Church 2007). The most merit humans can have is condign—when, under the impetus of God’s grace, they perform acts which please him and which he has promised to reward (Rom. 2:6–11, Gal. 6:6–10). Thus God’s grace and his promise form the foundation for all human merit

So, in conclusion, we have assurance of salvation due to Christ and we do positive actions under the impetus of God's grace