It appears that the eye for an eye Law (Levitcus 24:20) is punishing, by taking away something that we need. Whereas the Law to bless our enemy (Matthew 5:38-5:34) is like giving something good to make them happy. Like being kind to the evil one and lending to our enemies without expecting repayment (Luke 6:35). In (Matthew 5:17) Jesus said he came to fullfill the eye for an eye Law, right down to the smallest detail.
The OP speaks of fulfilling the "eye for an eye Law, right down to the smallest detail." That isn't quite right however. Jesus said "Do not think that I have come to abolish the Law or the Prophets; I have come not to abolish but to fulfill." He is speaking very generally here. The law of an "eye for an eye" needs to be seen in its context: In Exodus, it is an instruction to judges to award plaintiffs reasonable damages, no less and no more.
When people who are fighting injure a pregnant woman so that there is a miscarriage and yet no further harm follows, the one responsible shall be fined what the woman’s husband demands, paying as much as the judges determine. 23 If any harm follows, then you shall give life for life, 24 eye for eye, tooth for tooth, hand for hand, foot for foot, 25 burn for burn, wound for wound, stripe for stripe. (Ex. 21)
Indeed the very next verse shows that the law of "eye for eye" is NOT to be taken literally:
26 When a slaveowner strikes the eye of a male or female slave, destroying it, the owner shall let the slave go, a free person, to compensate for the eye. 27 If the owner knocks out a tooth of a male or female slave, the slave shall be let go, a free person, to compensate for the tooth.
Admittedly other iterations of the "eye for an eye" rule seem to imply a more literal sense of the matter (Dt 19:21, Lev 24:19). But they come on the foundation of Exodus' version and do not intend to supersede Exodus' insistence than monetary compensation ('paying as much as the judges determine') is an acceptable alternative to physical harm. There were lively debates in Jesus' time about how the Law was to be applied in many areas, and when Jesus said he came to "fulfill the law" it does not mean that he necessarily that the harshest interpretation of the law was correct.
Now that we understand that "eye for eye" need not be taken literally, here is an example of how seeking "eye-for-eye" compensation and loving one's enemy are compatible. Let's say a man runs a stop sign and causes a serious injury to the diver of the other car. The injured party sues for damages equal to the pain and loss from his injury, plus damages to the car. But spiritually, he also forgives the person who caused the injury and they become friends. His enemy in the realm of law has become his brother in spirit.
Twice in that passage Jesus says "But I say to you..." He definitely sees a difference between the law quoted in v38 and the advice given in vv39-42, and between the law quoted in v43 and the advice given in vv 44-48.
But in Matthew ch22 v39, Jesus declares that that "the law and the prophets" depend upon the two greatest commandments. So I suggest that those two commandments in themselves are "the law and the prophets" which he promises to keep in ch5 v17. Not the full detail of the laws of Moses.
That is how we may solve these "contradictions" about whether the N.T. does or does not support the law. By understanding that the word "law" has more than one meaning.