I think (with one caveat) that your question has a negative answer for families of very squashed spheres.
Let $$ D_a^\pm=\{(x,y,\pm a^{-1}): x^2+y^2\leq a\} $$ be the disk of radius $a$ at height $\pm a^{-1}$ and let $$ T_a= \{(x,y,z):(x-a \cos(\theta))^2+(y-a\sin (\theta))^2+z^2=a^{-2}, \theta\in [0,2\pi]\} $$ be the the surface obtained by rotating the semicircle of radius $a^{-1}$ centered at $(0,0, a)$ around the $z$-axis.
Set $$\Gamma_a=D_a^+\cup T_a \cup D_a^-.$$ This is a $C^{1,1}$ convex surface. Clearly as long as $a\geq 1$:
- The Gauss curvature is bounded between $0$ and $2$.
The Gauss curvature is bounded between $0$ and $2$.
- The maximum of the mean curvature is at least $a$.
The maximum of the mean curvature is at least $a$.
- The injectivity radius is bounded from below by $a$.
The injectivity radius is bounded from below by $a$.
- If desired these can be smoothed without changing the relevant properties.
If desired these can be smoothed without changing the relevant properties.
The convexity ensures this surface is isometrically rigid in $\mathbb{R}^3$ so there is no other isometric immersion into $\mathbb{R}^3$. Letting $a\to \infty$ shows that one can't have your desired estimate.
The convexity ensures this surface is isometrically rigid in $\mathbb{R}^3$ so there is no other isometric immersion into $\mathbb{R}^3$. Letting $a\to \infty$ shows that one can't have your desired estimate.
The caveat: I don't know whether one has (or should expect) this rigidity when one embeds into higher codimension.
