Were there any long-term political effects on Argentina after Eichmann's kidnapping?

In short, the answer is "no" to long term political effects. The article you refer to starts with a very good summation (and is my source throughout):

The contrast between the very speedy resolution of the crisis in Israeli-Argentine relations and the affair's long-lasting effects on Argentina's Jews indicates once again that the interests of the local Jewish community and those of the state of Israel--which defined itself on the day of its birth as "the Jewish state"--are not completely congruent and involve, at times, different dynamics. It also indicates that Argentine authorities were too often unwilling or unable to put a stop to antisemitic attacks by nationalist groups. Instead, they opted for closer relations with the Jewish state, hoping in this way to prevent such attacks from blackening Argentina's image in Western public opinion in general and in the American media in particular. Securing U.S. support and economic cooperation was, after all, a prime goal for all Argentine governments in the post-World War II era.
_{—Rein, 'The Eichmann Kidnapping: Its Effects on Argentine-Israeli Relations and the Local Jewish Community'}

Original Reaction

The original statements on Eichmann's capture did not reference a country, but media soon proclaimed it as 'Argentina'.

The Argentine foreign minister, Diógenes Taboada, promptly requested an unequivocal statement from Ambassador Levavi as to whether Eichmann had been arrested in Argentina. "If Eichmann was captured in Argentina, that is contrary to international norms and will compel Argentina, despite its good relations with Israel, to register a most serious protest, with unforeseeable consequences."

Ambassader Levavi advised his own government that the Argentines had to take such measures as the opposition was strong and they needed a show of force. The Israeli argument for denying the truth was based on maintaining the standing of the local Jews as well as Israel's own position in Latin America.

...[Argentine President Frondizi] found himself caught between opposing forces, "[on one hand] those who believed that Argentina should not press any claim, since it would mean defending a criminal like Eichmann, [and on the other hand] the pressure from those who wanted to turn the problem into a means of persecuting Jews." ...

To show he was taking a firm stand, Frondizi initially ordered the recall of Argentina's ambassador from Tel Aviv for consultations. However, by June 11, Levavi reported to the Israeli foreign ministry that Frondizi's Jewish adviser had told him: "The president has decided to end the dispute over Eichmann. He wants the affair to be submitted to the U.N. and to sink into the archives somewhere. No one intends for us to return Eichmann. ..." ...

UN Measures

The Argentine Ambassador to the UN, Mario Amadeo, was, however, far more unyielding in his approach to the question and met with Israeli representatives several times, demanding Eichmann's return. Amadeo's accomplishments in the UN culminated in a Security Council meeting:

...on June 22, 1960, to discuss Argentina's complaint. The Argentines demanded a debate on the infringement of their sovereignty and a condemnation of Israel for kidnapping Eichmann in violation of the rules of international law and the U.N. goals as expressed in its charter and conferences. The Security Council duly met, condemned Israel, and ordered it to give Argentina "appropriate reparations".

The US, the UK, and France voted for Argentina while Poland and the USSR abstained in support of Israel. Nevertheless, the final motion had also input a clause recommending that friendly relations be restored soon. Also, Argentina's status as an asylum country was quite well known for all types of political refugees (and was used as an argument by Amadeo in the UN).

Other Measures

Argentina made its most dramatic move on July 22, when it declared Levavi, Israel's ambassador, "persona non grata." ... Levavi ... was later to look back on the act as something Frondizi did against his will, in an attempt to preserve his image. A notable feature of the testimony provided by Israeli diplomats serving in Buenos Aires at the time is that they met with no angry or vengeful hostility on the part of the authorities during the two months of the diplomatic crisis.

By August, the worst was over. Argentines chose Levavi's removal as the best way to resolve the situation:

For the government of Argentina, the removal of Levavi will wipe out the incident and it will be possible to re-establish normal relations between the two states.

... the Argentine president also stressed that they had "decided to wipe out the incident and emphasized in particular economic motives connected with the development of the state. He already sensed a certain aloofness toward Argentina on the part of wealthy Jews around the world, and such an aloofness might disturb his plans."

Resolution

A one-paragraph communiqué was drafted jointly, including an apology by Israel for violating the sovereignty of Argentina along with a declaration that the incident had passed with the resumption of normal diplomatic relations. Further improvements were demonstrated in September 1960, when Argentine ministers visited an Israeli archaeological exhibition in Buenos Aires. Further,

Even before the end of the year, the two states had exchanged new ambassadors who promptly announced that relations had returned to the same friendly level as before the crisis.

Thus, the initiative for ending the crisis ... came from the Argentines and was not the result of negotiations between the two states. The improvement ... must be viewed ... in the context of Frondizi's strong desire to strengthen Argentina's political and economic ties with the United States. ... Frondizi had been the first Argentine president to visit the United States... [Frondizi] did not want to appear indifferent to the defense of Argentina's national sovereignty, but he recognized—perhaps even overestimated—the influence and economic power of the U.S. Jewish community, and he wanted to avoid unnecessary clashes with American public opinion over the Eichmann affair.