The order of events happening in the cell is that the damaged DNA is modified, halting further transcription. Repair of the damaged DNA then entails cutting out the modified DNA, which is then secreted. That secreted 8-OHdG is what the kit then detects. So, the secreted sample really represents the conclusion of the process, while testing inside cells would represent the beginning. The DNA in tissue could theoretically contain modified bases, but whether or not this kit would be able to detect them is another matter. Going through the literature, there is one research article where this kit was used to determine the levels of DNA damage in cell lysates (Lee et al., 2010). You might be able to gather further information regarding the preparations of cell lysates from there.