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An equivalent approach is to convert everything in terms of hours first before we begin calculations of time.

$800\textrm{ hrs }480\textrm{ min }5400\textrm{ sec }= 809.5\textrm{ hrs }\equiv (-6.5)\textrm{ hrs}\pmod{24}$

As we're working backward in time, we add the absolute value of the residue obtained to our current time and then add the $1010\textrm{ hrs}\bmod 24$ to obtain time of telecast as follows:

$4:00\textrm{ PM}\stackrel{+6.5\textrm{ hrs}}\longrightarrow 10:30\textrm{ PM}\stackrel{+(1010\textrm{ hrs})~\bmod~24}\longrightarrow 12:30\textrm{ AM}$

Now, comes the date. We first have $809.5=816-6.5$ . Since $4:00$ PM and $10:30$ PM comes in a single day after $6.5$ hrs, we have the day of last episode as $\frac{816}{24}=34$ days before current day. Working from current day, we have,

$34\textrm{ days }=\left(\underset{\textrm{index on March }1}{2}\right)+\left(\underset{\textrm{index on Feb }1}{29}\right)+\left(\underset{\textrm{index on Jan }1}{31}\right)-\left(\underset{\textrm{current index to final day}}{28}\right)$

Hence, the last telecast was on $28^{\textrm{th}}$ Jan $^\prime2015$ at $10:30\textrm{ PM}$ . Now, we calculate day-hour increments after $1010\textrm{ hrs}$ as follows:

$1010\textrm{ hrs }=\frac{1008}{24}\textrm{ days }+2\textrm{ hrs }=42\textrm{ days }+2\textrm{ hrs}$

Working from day of last telecast, we have,

$42\textrm{ days }=\left(\underset{\textrm{going to Feb }1}{3}\right)+\left(\underset{\textrm{going to March }1}{29}\right)+\left(\underset{\textrm{date reached}}{10}\right)$

This gives us $10:30\textrm{ PM }$ of $10^{\textrm{th}}$ March $^\prime2015$ . We still have to add the remaining $2$ hrs. Since we have new day after $12:00\textrm{ AM}$ and the final time we get for next telecast is $12:30\textrm{ AM}$ , we add another day to our obtained date.

This gives us the date of $11^{\textrm{th}}$ March $^\prime2015$ and time of $12:30\textrm{ AM}$ as date and time of next telecast.

By the way, there's are minor typos in few lines of your solution (mostly $\LaTeX$ problems). I suggest you edit them properly to make your solution cleaner.