How are John 18:36 and Matthew 13:41 reconciled according to Protestants? John 18:36:  


  My kingdom is not of this world. If it were, my servants would fight to prevent my arrest by the Jewish leaders. But now my kingdom is from another place.  


Matthew 13:41:  


  The Son of Man will send out his angels, and they will weed out of his kingdom everything that causes sin and all who do evil.


In John he states that his kingdom is not of this world but in Matthew he talks about his angels weeding out of his kingdom all who do evil. So if his kingdom is not of this world, how are these two verses reconciled?
This can be reconciled, like many other apparent contradictions by understanding the context of the passages.  See Rules behind resolving alleged Biblical discrepancies.


  ...the vast majority of "contradictions" can be attributed to one of
  two possible causes.
  
  
  Copyist errors  
  A misunderstanding of the context. (Historical context, cultural context, taking a verse out of context with related passages of
  Scripture, different meanings for words used at the time a translation
  was written compared to the meaning of the words today, etc.)
  


The passage from John, as covered in many commentaries is in the context of an accusation that Jesus is some threat to the existing rule.  Sample:


  Barnes' Notes on the Bible
  
  My kingdom ... - The charge on which Jesus was arraigned was that of
  laying claim to the office of a king. He here substantially admits
  that he did claim to be a king, but not in the sense in which the Jews
  understood it. They charged him with attempting to set up an earthly
  kingdom, and of exciting sedition against Caesar. In reply to this,
  Jesus says that his kingdom is not of this world - that is, it is not
  of the same nature as earthly kingdoms. It was not originated for the
  same purpose, or conducted on the same plan. He immediately adds a
  circumstance in which they differ. The kingdoms of the world are
  defended by arms; they maintain armies and engage in wars. If the
  kingdom of Jesus had been of this kind, he would have excited the
  multitudes that followed him to prepare for battle. He would have
  armed the hosts that attended him to Jerusalem. He would not have been
  alone and unarmed in the garden of Gethsemane. But though he was a
  king, yet his dominion was over the heart, subduing evil passions and
  corrupt desires, and bringing the soul to the love of peace and unity.
  
  Not from hence - That is, not from this world.


The passage from Matthew, on the other hand, is part of a parable where he is using the sample of weeds and wheat in the same field to illustrate that righteous and evil will be living together on this earth.  He is referring to His spiritual Kingdom, or the Church. Example from another commentary: (Emphasis added by me)


  Gill's Exposition of the Entire Bible
  
  The Son of man shall send forth his angels,.... Meaning himself, whose
  ministers the angels are; who wait upon him, and are at his beck and
  command; even the thousand thousands that minister unto him; these
  will be sent forth by his orders, into the several parts of the world,
  where he has any churches, or an interest,
  
  and they shall gather out of his kingdom: the Gospel church, over
  which Christ is king, where he rules and governs in the hearts of his
  people; and who are cheerfully and willingly obedient to his laws,
  under the influence of his Spirit and grace: but all who are in the
  visible Gospel church state, are not such; some are wicked and
  rebellious, and though they are suffered to continue, yet not always;
  for if not removed by censures and excommunications, they will be at
  last by angels; who will separate them from the saints:
  
  even all things that offend; who are scandals to Christ, his church,
  and Gospel, by their wicked principles, or infamous practices; and who
  give offence, not only to God, and his righteous law, but lay
  stumbling blocks in the way of the children of God, and are the
  authors of divisions and offences among them:
  
  and them that do iniquity; that do nothing else but iniquity; and who,
  though they profess to be religious persons, are secretly, or openly,
  workers of iniquity; and are even doing iniquity, in and whilst they
  are professing religion.


As with almost any word in the English language, the context helps to define the meaning of the word, so when taken in context, the apparent discrepancy goes away.