Delete ApexLogs where the LogUser returns null, causing a JS error. Run this in the browsers javascript console when on the Developer Console

DeleteLogsWithNullLogUsers.js

SfdcDevConsole.ToolingAPI.query('SELECT Id, LogUserId, LogUser.Name FROM ApexLog', {
    continuation : function(response) {
        var records = response.records;
        console.log('Found ' + records.length + ' log(s)');
        var idsToDelete = [];
        for (var i = 0; i < records.length; i++) {
            var rec = records[i];
            console.log('\t' + rec.Id + ' ' + rec.LogUserId);
            if (!rec.LogUser || rec.LogUser === 'null') {
                idsToDelete.push(rec.Id);
            }
        }
        
        function removeLog(i, logId) {
            //Stagger so we don't send all the request at once.
            setTimeout(function() {
                console.log('\tDeleting log ' + logId);
                SfdcDevConsole.ToolingAPI.DELETE('ApexLog', logId, {});
            }, 500*i);
        }
        
        console.log('Deleting ' + idsToDelete.length + ' log(s)');
        //Should probably use bulk api. O-well... just delete one at a time
        for (i = 0; i < idsToDelete.length; i++) {
            removeLog(i, idsToDelete[i]);
        }
    }
});