LiOH hydrolysis of methyl 2,2-dimethoxyacetate not giving product?
If your reaction has worked and you get the expected product, you should see following NMR data as the sought literature reported:

$\mathrm{^{1}H}$-NMR: $\delta$ 3.44 (s, $\ce{6H}$), 4.85 (s, $\ce{1H}$), and 10.32 (broad s, $\ce{1H}$).
$\mathrm{^{13}C}$-NMR: $\delta$ 54.1 ($\ce{2CH3}$), 98.6 ($\ce{CH}$), and 170.7 ($\ce{C_q}$).

Instead, you got two peaks, $\delta$ 3.704 (s, $\ce{6H}$) and $\delta$ 3.458 (s, $\ce{1H}$), in your $\mathrm{^{1}H}$-NMR. And, you got only one peak at $\delta$ 67.12 in your $\mathrm{^{13}C}$-NMR. All this data telling me is your reaction has not worked as expected. What you have is 1,4-dioxane during the ether extract, $\mathrm{^{1}H}$-NMR of which gives only a singlet at $\delta$ 3.71 when $\ce{CDCl3}$ is the solvent. It also gives one peak at $\delta$ 67.14 in $\mathrm{^{13}C}$-NMR under same solvent (Ref.1).
I believe your compound may still be in the aqueous phase. You may need to saturate the aqueous phase with salt ($\ce{NaCl}$) before extract the product to organic phase. Note that acetic acid is very soluble in water. Therefore, I expect 2,2-dimethoxyacetic acid to be highly soluble in water as well (a $\ce{C4(H2O)4}$ compound like a monosaccharide).
References:

H. E. Gottlieb, V. Kotlyar, A. Nudelman, “NMR Chemical Shifts of Common Laboratory Solvents as Trace Impurities,” J. Org. Chem. 1997, 62(21), 7512–7515 (DOI: 10.1021/jo971176v).