Where does the word 'Eloheim' originate from? Elohim being referred to as the Father whom is a God 


  When one speaks of God, it is generally the Father who is referred to; that is, Elohim. - God (The Church of Jesus Christ of Latter-Day Saints).


Eloheim associates with multiple 'Gods' in this particular article

Enrichment Section A: Who Is the God of the Old Testament (The Church of Jesus Christ of Latter-Day Saints)


  The word Elohim is a plural form of the Hebrew word for God, although
  modern scholars agree that it should be taken as a singular noun even
  though the im ending is a plural form. Joseph Smith, however,
  indicated the significance of the plural form:
  
  “If we pursue the Hebrew text further, it reads, … ‘The head one of
  the Gods said, Let us make a man in our own image.’ I once asked a
  learned Jew, ‘If the Hebrew language compels us to render all words
  ending in heim in the plural, why not render the first Eloheim
  plural?’ He replied, ‘That is the rule with few exceptions; but in
  this case it would ruin the Bible.’ He acknowledged I was right.
  
  “In the very beginning the Bible shows there is a plurality of Gods
  beyond the power of refutation. It is a great subject I am dwelling
  on. The word Eloheim ought to be in the plural all the way
  through—Gods. The heads of the Gods appointed one God for us; and when
  you take [that] view of the subject, it sets one free to see all the
  beauty, holiness and perfection of the Gods.” (Teachings, p. 372.)


In this specific case Joseph Smith is referring to Genesis where it says


  'Let us make man in our own image' - Genesis 1:26


So it is evident that he is using Genesis as the core of the word 'Eloheim'. In the original Hebrew manuscripts; the writer uses the word Elohim 


  ’ĕ·lō·hîm (אֱלֹהִ֖ים)


From my research it seems 'Eloheim' doesn't exist in the Hebrew nor English vocabulary (I could be very wrong, so please correct me if so). Now let us consider if this was a typo - Why would the Father be referred to the same as the plurality of Gods? 

Also note how the 'learned Jew' said it would ruin the Bible, so it is evident he is talking about the Bible in this context.

With all this in mind, where does the word 'Eloheim' originate from?
There seem to be two basic questions here:


  In the original Hebrew manuscripts; the writer uses the word Elohim....From my research it seems 'Eloheim' doesn't exist in the Hebrew nor English vocabulary.


Of course, in the "original Hebrew" manuscripts, no vowels were written, so technically no distinction could be drawn between these two forms. On the other hand, in the Masoretic Text (to which vowel pointings were added in the Middle Ages, and on which all modern translations are substantially based) I can confirm that there is no word that would be transliterated eloheim. In fact, the only word with the consonants אלהים is pointed every time as the OP indicated: אֱלֹהִים (ʾelohîm). 

While transliteration schemes differ, neither the Wikipedia article on Romanization nor the SBL guidelines suggest that chireq-yod may be transliterated ei (rather: iy or î). However, as noted in a comment, English spelling was not standardized at the time Joseph Smith wrote this, much less transliteration from Hebrew. This is exemplified in this collection of Smith's writings where Genesis 1:1 is given:


  בְּרֵאשִׁ֖ית בָּרָ֣א אֱלֹהִ֑ים אֵ֥ת הַשָּׁמַ֖יִם וְאֵ֥ת הָאָֽרֶץ
  Berosheit baurau Eloheim ait aushamayeen vehau auraits (Smith)
  Bereʾshit baraʾ ʾelohim ʾet hashshamayim weʾet haʾarets (simplified modern transliteration)


There are numerous letter choices in there which don't make sense to me (mostly vowels), but he was obviously working with different standards. 

OP further asks:


  Why would the Father be referred to the same as the plurality of Gods?


This is a vexed question about which no few books have been written. The basic answer is that this is just a quirk of how the Hebrew language works. A few supporting points:


This word is not unique. Other words translated as singular include חִטִּים (chittim, "wheat"), מַ֫יִם (mayim, "water"), שָׁמַ֫יִם (shamayim, "sky"), and תַּנִּינִים (tanninim, "dragon"), among others. As one respected Hebrew grammar puts it (p. 468):


  [The plural] used, quite broadly, to designate things which, while having a real unity, also express plurality in some way. Thus in a compound object one may consider the component elements, in an extended object the various parts, in a particularly perfect being the multiplicity or the intensity of the being, or even in something abstract the multiplicity of manifestations. Thus in Hebrew one may distinguish the plurals of composition, of extension, of excellence or of majesty, of intensity, and of abstraction.


The authors include ʾelohîm as a "plural of excellence or majesty". See also the older but still valuable discussion by Gesenius on the topic.
Although plural in form, ʾelohîm, when it unambiguously refers to the God of Israel, is consistently used with singular verbs. Pronouns pointing to ʾelohîm are also consistently singular.1 This is a strong basis on which to tranlate it as singular in English.
The OP quotes an LDS source suggesting that the term should be rendered "Gods". This is based on the false premise that "the Hebrew language compels us to render all words ending in heim in the plural".    

On the other hand, occasionally trinitarians are heard making claims about the plurality of the Godhead based on morphology of this word. From a linguistic and textual perspective, I am aware of no basis for this, and the phenomenon is adequately explained as above. 





1 There are rare exceptions, one of which is quoted in the question: Gen 1:26. This text deserves special attention; see the Q&amp;A on Hermeneutics.SE. For the more general issue, see also Why is Elohim translated as God rather than gods in Genesis 1:1?, which covers much of the same ground as this one.