Simplifying the following mathematical expression using a computer?

Here's an all-form-and-no-substance simplification that merely introduces the notation $$X_p:= frac{v^p-v^{-p}}{t^p-t^{-p}}$$ and factors-out powers of $t$ and $v$ from certain groups:

$$begin{align} W_{(1,1)}(t,v)&=frac{t^{-2k}v^k}{12}left( -4X_{3/2}X_{1/2}+3X_1^2+X_{1/2}^4right) \[4pt] &+frac{t^{-k} v^k}{8}left(-2X_2 + X_1^2+2X_1X_{1/2}^2-X_{1/2}^4right) \[4pt] &+frac{t^k v^k}{24}left(-6X_2 +8X_{3/2}X_{1/2}+3X_1^2-6X_1X_{1/2}^2+X_{1/2}^4right) end{align} tag{1}$$

That the $X$-subscripts in each term (interpreting exponents as multipliers) sum to $2$ serves as a nice sanity-check. (OP's original rendering of the expression omitted the exponent on the $X_{1/2}$ term in the first line. The subscript check helped identify this error. Check the edit history for a version of this answer that included consideration of the erroneous expression.)

Dropping $(1)$ into Mathematica makes a bit of a mess. To de-messify it slightly, define $$s := sqrt{t} qquad u := sqrt{v}$$ so that half-powers of $t$ and $v$ become integer powers of $s$ and $u$. Then, simplifying $(1)$ grouping-by-grouping gives $$begin{align} W_{(1,1)}(s^2,u^2) = &-s^{-4k+6}u^{2k-4} frac{(u^2-1)^2 (s^2-u^2) (s^2 u^2-1)}{(s^2-1) (s^4-1)^2 (s^6-1)} \[4pt] &-s^{-2k+4}u^{2k-4}frac{left(u^2-1right) left(s^2-u^2right) left(s^2 u^2-1right)(s^4-u^2)}{(s^2-1)^2(s^4-1) (s^8-1)} \[4pt] &-s^{2k+4} u^{2k-4}frac{(u^2-1)(s^2-u^2)(s^4-u^2)(s^6-u^2)}{(s^2-1) (s^4-1) (s^6-1) (s^8-1)} \[4pt] =&phantom{-}frac{u^{2k}}{s^{4k}};frac{u_1^2 m_1 m_{-1}}{s_1 s_2^2 s_3} +frac{u^{2k}}{s^{2k}};frac{u_1 m_1 m_2m_{-1} }{s_1^2 s_2 s_4} -s^{2k}u^{2k};frac{u_1 m_1 m_2 m_3}{s_1 s_2 s_3 s_4} \[4pt] =&phantom{-}frac{u^{2k}}{s^{4k}};frac{ u_1m_1(s_1 s_4 u_1 m_{-1} +s^{2k}s_2 s_3 m_2 m_{-1} -s^{6k}s_1 s_2 m_2 m_3) }{s_1^2s_2^2s_3 s_4} end{align}$$ where $$s_p:=s^p-s^{-p} qquad u_p:=u^p-u^{-p} qquad m_p := frac{s^p}{u}-frac{u}{s^p}$$

Further defining $r_p := s^p+s^{-p}$, we have $$begin{align} s_3 &= s^3-s^{-3}= (s-s^{-1})(s^2+1+s^{-2}) = s_1 (r_2+1) \ s_4 &= s^4-s^{-4}= (s^2-s^{-2})(s^2+s^{-2}) = s_2 r_2 end{align}$$ which allows us to write

$$W_{(1,1)}(s^2,u^2)=frac{u^{2k}}{s^{4k}};frac{ u_1m_1(r_2 u_1 m_{-1} +s^{2k} (r_2+1) m_2 m_{-1} -s^{6k} m_2 m_3) }{s_1 s_2 s_3 s_4} $$

There may be even cleaner ways to write the expression, but this is as far as I'll go.

Wolfram|Alpha supports Latex. Unfortunately, the engine didn't seem to understand your query, perhaps because of its length. If possible, I would suggest splitting the expression up into chunks, and entering them one by one into Wolfram|Alpha.