Isn't there any divisor $k$ of $n^4$ such that $n^2-n<k<n^2$?

Let $0<i<n$ and suppose that $n^2-i$ divides $n^4$. Then, $$ n^4=(n^2-i)(n^2+j)=n^4+(j-i)n^2-i\,j\implies i\,j=(j-i)n^2 $$ for some integer $j$. If $j\le i$, then $$ n^4=(n^2-i)(n^2+j)\le(n^2-i)(n^2+i)=n^4-i^2. $$ Thus, $j>i$. Since $i<n$, it follows that $j>(j-i)n\ge n$. Moreover $$ n\le j=\frac{i\,n^2}{n^2-i}<\frac{i\,n^2}{n^2-n}=\frac{i\,n}{n-1}\implies i>n-1, $$ a contradiction with the fact that $i<n$.

Answer to the Question

Let $k=n^2-a$. Since $n^2-a\mid n^4$ and $$ \frac{n^4}{n^2-a}=n^2+a+\frac{a^2}{n^2-a} $$ we must have $d=\frac{a^2}{n^2-a}\in\mathbb{Z}$. However, since $\color{#C00}{1\le a\le n-1}$ and $\frac{a^2}{n^2-a}$ is increasing in $a$, $$ 0\lt\overbrace{\ \frac1{n^2-1}\ }^{\large\color{#C00}{a=1}}\le d\le\overbrace{\frac{n^2-2n+1}{n^2-n+1}}^{\large\color{#C00}{a=n-1}}\lt1 $$ which is impossible because there is no integer between $0$ and $1$.

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Bounding the Greatest Factor of $\boldsymbol{n^4}$ less than $\boldsymbol{n^2}$

Case: $\boldsymbol{d=1}$

Since $n^2=a^2+a$, we have $n-1\lt a\lt n$, which is impossible.

Case: $\boldsymbol{d=2}$

Since $n^2=\frac{a(a+2)}2$, $\sqrt2\,n-1\lt a\lt\sqrt2\,n$. In fact, $2n^2+1=(a+1)^2$. That means that $$ \left(\frac{a+1}n\right)^2=2+\frac1{n^2}\lt\left(\sqrt2+\frac1{2n^2}\right)^2 $$ Which forces $\frac{a+1}n$ to be a continued fraction overestimate of $\sqrt2$ . This gives the first pairs $(n,a)$ to be $$ \{(0,0),(2,2),(12,16),(70,98),(408,576),(2378,3362),(13860,19600)\} $$ The recursion for $n_k$ is $n_k=6n_{k-1}-n_{k-2}$ and $a_k=\left\lfloor\sqrt2\,n\right\rfloor$.

In any case, we have

The largest factor of $n^4$ less than $n^2$ must be less than $n^2-\sqrt2\,n+1$ and there are an infinite number of $n$ so that the largest factor of $n^4$ less than $n^2$ is greater than $n^2-\sqrt2\,n$.

The best examples come from $n=2xy,$ where $x,y>0$ are integers and $x^2 - 2 y^2 = \pm 1 \; . \;$ In these cases we get a factor $m$ of $n^4$ near $n^2 - n \sqrt 2 \; . \; $ When $x^2 - 2 y^2 = 1,$ we get $m = n^2 - 2 x^2.$ When $x^2 - 2 y^2 = -1,$ we get $m = n^2 - 4 y^2.$

jagy@phobeusjunior:~$ ./mse
 n: 2 =  2  m:  2  n^2 - m:  2 =  2
 n: 3 =  3  m:  3  n^2 - m:  6 =  2 3
 n: 4 =  2^2  m:  8  n^2 - m:  8 =  2^3
 n: 6 =  2 3  m:  27  n^2 - m:  9 =  3^2
 n: 10 =  2 5  m:  80  n^2 - m:  20 =  2^2 5
 n: 12 =  2^2 3  m:  128  n^2 - m:  16 =  2^4
 n: 24 =  2^3 3  m:  512  n^2 - m:  64 =  2^6
 n: 30 =  2 3 5  m:  810  n^2 - m:  90 =  2 3^2 5
 n: 60 =  2^2 3 5  m:  3456  n^2 - m:  144 =  2^4 3^2
 n: 70 =  2 5 7  m:  4802  n^2 - m:  98 =  2 7^2
 n: 84 =  2^2 3 7  m:  6912  n^2 - m:  144 =  2^4 3^2
 n: 140 =  2^2 5 7  m:  19208  n^2 - m:  392 =  2^3 7^2
 n: 168 =  2^3 3 7  m:  27783  n^2 - m:  441 =  3^2 7^2
 n: 180 =  2^2 3^2 5  m:  32000  n^2 - m:  400 =  2^4 5^2
 n: 408 =  2^3 3 17  m:  165888  n^2 - m:  576 =  2^6 3^2
 n: 594 =  2 3^3 11  m:  351384  n^2 - m:  1452 =  2^2 3 11^2
 n: 816 =  2^4 3 17  m:  663552  n^2 - m:  2304 =  2^8 3^2
 n: 1170 =  2 3^2 5 13  m:  1366875  n^2 - m:  2025 =  3^4 5^2
 n: 2378 =  2 29 41  m:  5651522  n^2 - m:  3362 =  2 41^2
 n: 3230 =  2 5 17 19  m:  10425680  n^2 - m:  7220 =  2^2 5 19^2
 n: 4756 =  2^2 29 41  m:  22606088  n^2 - m:  13448 =  2^3 41^2
 n: 5880 =  2^3 3 5 7^2  m:  34560000  n^2 - m:  14400 =  2^6 3^2 5^2
 n: 13860 =  2^2 3^2 5 7 11  m:  192080000  n^2 - m:  19600 =  2^4 5^2 7^2
 n: 16296 =  2^3 3 7 97  m:  265531392  n^2 - m:  28224 =  2^6 3^2 7^2
 n: 27720 =  2^3 3^2 5 7 11  m:  768320000  n^2 - m:  78400 =  2^6 5^2 7^2
 n: 42672 =  2^4 3 7 127  m:  1820786688  n^2 - m:  112896 =  2^8 3^2 7^2
 n: 57960 =  2^3 3^2 5 7 23  m:  3359232000  n^2 - m:  129600 =  2^6 3^4 5^2
 n: 58206 =  2 3 89 109  m:  3387795864  n^2 - m:  142572 =  2^2 3 109^2
 n: 80782 =  2 13^2 239  m:  6525617282  n^2 - m:  114242 =  2 239^2