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During tunnel construction, if the sinking rate of the tunnel arch is too fast, it may compromise the safety of the construction project, and adjustments to the support parameters need to be made promptly. A construction team is monitoring the sinking amount of the Xiangshan Tunnel project in Fuzhou. By conducting a regression analysis on the monitoring results, a regression equation is established between the cumulative total sinking amount $z$ (unit: millimeters) of the tunnel arch in the previous $t$ days (unit: days). Based on this regression equation, predictions can be made to determine if adjustments to the support parameters are necessary. The measured data of the tunnel arch sinking is as follows:

| $t$ (days) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|------------|---|---|---|---|---|---|---|
| $z$ (mm)   | 0.01 | 0.04 | 0.14 | 0.52 | 1.38 | 2.31 | 4.30 |

Researchers created a corresponding scatter plot and observed that the function $z=ke^{bt}$ could be used for fitting. Let $u=\ln z$, and the following calculations were obtained:

$\overline{z}=1.2$, $\sum_{i=1}^{7}({t}_{i}-\overline{t})({z}_{i}-\overline{z})=22.3$, $\sum_{i=1}^{7}({z}_{i}-\overline{z})^{2}=27.5$, $\overline{z}=1.2$, $\sum_{i=1}^{7}({t}_{i}-\overline{t})({u}_{i}-\overline{u})=25.2$, $\sum_{i=1}^{7}({u}_{i}-\overline{u})^{2}=30$.

$(1)$ Try to establish the regression equation between $z$ and $t$ and predict the cumulative total sinking amount of the tunnel arch in the first 8 days (rounded to 0.1).

$(2)$ It is known that when the instantaneous sinking rate of the arch exceeds 27 mm/day at a certain moment, the support system will be overloaded, and there is a risk of tunnel collapse. The construction team needs to adjust the support parameters one day in advance. Estimate the latest day when the support parameters need to be adjusted (rounded to the nearest integer).

Given: 
- Regression line formula: $\hat{y}=\hat{a}+\hat{b}x$
- Slope and intercept estimation formulas: $\hat{b}=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}({x}_{i}-\overline{x})^{2}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$
- Reference data: $\sqrt{210}≈14.5, \ln 10≈2.3, \ln 30≈3.4, {e}^{24}≈11.0, {e}^{2}≈7.38$.

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