I recommend storing the plots in a table, not as separate definitions of k[1,1] k[1,2] etc., so you can take advantage from parallelization.
Further, solve the equation for each i-j-pair first and use it in the Plot-command.
But the gain of speed vanishes when using greater precision.
Please have a look at the attachment (most graphics deleted to keep the notebook small) to see the difference.
Attachment: DiffeqTable.nb, URL: ,