What made Elizabeth I a good leader? What do you think was the most challenging issue she faced, and how did she overcome it?

Queen Elizabeth I was possessed of many positive character traits. She was highly intelligent, single-minded, and intensely practical, making compromises as and when required. When she died in 1603, England was undoubtedly a much stronger, more prosperous, and more respected country than it was when she ascended the throne.
Arguably the most challenging issue she faced was that of religion. In the 16th century, religion was far from being merely a matter of private belief; it was an issue of great national importance for the peace and security of the state. Elizabeth had succeeded her half-sister Mary, a deeply devout Catholic who had taken England back to full communion with Rome. Elizabeth was herself a Protestant, and it was clear to all that England under her rule would revert to being a Protestant country.
However, how she carried out this process of reconversion was significant and displayed her enormous political talents. Although Elizabeth was indeed a sincere Protestant, she was no fanatic. She saw the issue of the Church of England in fundamentally political, rather than religious, terms. This meant that the Church as an institution was there to support the monarchy as essentially a department of the state. In reforming the Church, Elizabeth set about building an institution that would greatly add to the stability of her throne while, at the same time, bringing some measure of peace to the realm.
How, then, did Elizabeth achieve this aim? The Settlement of 1559 was an attempt to compromise between rival factions within the Church of England. The Puritans were those Protestants who looked to Calvin's Geneva for inspiration. They wanted to see the establishment of a "Godly state" in which the secular authorities were thoroughly Christianized. Elizabeth was strongly opposed to the merest hint of theocracy; this would undermine the power and authority that had so nearly been taken from her under Mary.
There were also Anglicans who saw the Church as continuing the traditions of the Holy Catholic and Apostolic Church, though without what they saw as Rome's serious theological and structural errors. Elizabeth was not particularly sympathetic to this particular wing of the Church, although she did retain a fondness for certain remnants of Catholic worship, such as the use of vestments in services at the Royal Chapel.
Elizabeth was much less interested in the niceties of theology than in maintaining and strengthening royal supremacy. This is what mattered to her more than anything else; this principle was at the heart of the 1559 Settlement. Elizabeth made herself Supreme Governor of the Church of England. All those within the Church were now legally required to take an oath of loyalty to the monarch.
What the Settlement attempted to do was to accommodate as many differences of faith as possible within the Church. It was hoped by Elizabeth that this would isolate the fanatics of both extremes and make religious disputation much less bitter and divisive.
In due course, the 1559 Religious Settlement proved to be an uneasy political compromise, one whose unresolved tensions would ultimately lead to civil war nearly a century later. Although, in retrospect, it was probably the only arrangement that could have stabilized the country after nearly a quarter of a century of bitter religious conflict. However, only someone imbued with Elizabeth's strong sense of purpose and single-mindedness could have driven such a process to success. Several such successes would follow throughout her long reign, but the foundations had already been laid by the Religious Settlement of 1559, an act of leadership which greatly enhanced her reputation and authority in the eyes of her subjects and also of posterity.